Pietraszko’s Cave is a short story from the series Incredible Adventures of Beta and Bit.

Who are Beta and Bit?

They are brother and sister who constantly get into troubles. However, thanks to Bit’s hacking skills and Beta’s mathematical talent, they always manage to wriggle out of the difficulties.

Methods used by Beta and Bit are not fanciful creations of the authors – they are real tools employed by real data analysts. In this story Beta and Bit are going to use linear regression. The epilogue contains more information about that particular method.

This story was written by Magda Chudzian (plot) and Przemysław Biecek (statistics and design). Illustrations were created by Klaudia Korniluk. Translation into English was prepared by Katarzyna Pap, translation into Russian was prepared by Andrei Shpakov. The story was published by Foundation SmarterPoland.pl, ISBN: 978-83-65291-02-8. You will find more stories from this series at http://SmarterPoland.pl/BetaBit/

Are you ready to step into the world of Beta and Bit?

Chapter 1

- Dad’ll bite your head off. Or your fingers. And definitely he’ll take your tablet. You’ll be grounded –that’s a dead cert.

Bit barely heard what Beta said behind his back. He didn’t even notice when she entered the room. He’d just hacked into Proton, server with accounts users from Warsaw University of Technology. He had not expected to find anything interesting there really. He was wrong.

- What?

- Dad’ll bite your head off. You hacked into server at his work. Have you lost your marbles?

Well, that wasn’t wise to hack the server with user accounts... To be more specific, into the accounts of two employees of the Warsaw University of Technology. Dad’s documents included only some outlines of lectures and articles. But Bit opened also another account –he wondered what that freak Pietraszko might keep there.

- Chill out. He won’t find out. He also won’t find out that I plan to check what is that room you can see at the plan that shouldn’t be there. And why it’s walls are thicker than other walls. Will you join me?

- What do you think you’ll discover there? Top-secret arsenal? Artificial intelligence? You've got to be kidding me!

- Will you join me or not?

- Of course I will not leave you. I cannot really imagine that you can come up with something more stupid than what you’ve just done. But you’ve shown me many times how limited is my imagination.

They reached the University of Technology quite fast. They took the tram although the distance from the metro station to main building wasn’t great. They didn’t want to lose time on walking. The historical building always impressed them. The monumental facade was an epitome of science with a capital S. Now, however, their destination was a building of the Faculty of Electronics and Information Technology called ELKA located opposite the main building of the University of Technology and presenting a completely opposite architectural style. It was a gigantic rectangular block from a different epoch similar to thousands of other such building in Poland. A real labyrinth of corridors. Yet Beta and Bit knew it very well. They’ve spent many hours at the faculty as their father was a lecturer there and he took the kids with him to his work many many times.

They crossed Nowowiejska Street and went to one of the identical staircases protruding from the uniform shape of the building. Facade of each of the staircases was decorated with a mosaic. Beta and Bit slowed down as usual to look at them for a moment. They liked to guess what the mosaics could present. Once again they had the impression that the building of the faculty and its surroundings clashed. The huge geometrical building from 1960s stood not only next to the main building of the WUT from the previous century but also close to the pre-war tenement houses at Lwowska and Polna Streets and unique Warsaw Water Filters from the 19th century. It also took only a short walk along the Polna Street to reach the Łazienkowska Thoroughfare, a flagship investment of the People‘s Republic of Poland, which divided that historical street into two parts. A strange place indeed.

- Ooo, you came to your dad?

Bit silently passed Mrs. Ela. He run up the stairs. Beta smiled apologetically feeling ashamed of her brother. The charwoman had known them for a long time. She often talked to them when they were hanging around the faculty as small kids.

- Yes, for a second – the teenager responded politely. – I’ll just...

Mrs. Ela just nodded her head and Beta run after her brother.

- Wait. Stop – she tried not to shout. – What are you doing?

- You know I’m looking for... – he started to explain but Beta interrupted.

- You run like crazy. You draw attention to yourself. Remember that everybody here knows you. If we are to check something here really act normally, please.

- All right, sorry. I copied all the files, mainly some old photos and that plan. The room should be somewhere here, at the first floor.

Beta took a look at the tablet and sighed. – Your secret room overlaps with the room belonging to...

- Pietraszko – Bit finished with a smile. – How do you think, whose account I hacked to?

- Zięba – somebody shouted from behind the slightly ajar door to the laboratory. None of the several students standing in the corridor moved.

- Zięba! – the voice was louder and more threatening this time. Only then Igor Zięba left the wall he was supporting and moved towards the room.

- Sit down.

He sat down obediently. At the other side of the table there was a man wearing enormous glasses. His graying beard was hanging almost to his waist. Mr. Sławomir Pietraszko, MSc. A living proof of the fact that the academic advancement does not necessarily have to coincide with the size of academic achievements. His glasses might have remembered his student years. His beard remembered them for sure. Its history was almost an integral element of the curriculum for the first-year students. Igor Zięba knew it too. He evoked that story while he was waiting for the old Mr. Pietraszko, MSc, to start the exam. The story went like this:

Long, long time ago, but still within the memory of the oldest lecturers of the University, there was a student called Pietraszko. He was a very clever student and being still a young academic he devoted his career to the studies of the time-space continuum. He used the chaos theory to carry out simulations of the alternative evolutions of the history of the world. He was looking for relationships between possible scenarios. He was convinced that he discovered some universal law determining what may happen and what may never happen. Unfortunately, nobody believed in the seriousness and reasonableness of these studies. Pietraszko was offended and he stopped shaving in protest. He declared that he would shave again when somebody undermine results of his studies. In this way his 30-year-old beard originated. It would probably still grow for many years to come. Although everybody knew the history of the beard, nobody remembered what the young Pietraszko had tried to prove.

Igor Zięba waited patiently. Mr. Pietraszko, MSc with his nose in some papers did not look at the student at all. His beard was partially hidden under the desk. But his hair... His thin black and grey hair, oily as always, was hanging almost to his elbows. And that place: Pietraszko’s cave, his closed soundproof laboratory where the eccentric Master of Science closed himself everyday to avoid distractions and disturbance. Was there really no other unoccupied room?

- Are you prepared for the exam, rascal?

Zięba woke up from his meditations immediately. Mr. Pietraszko no longer looked at his notes. Now he petrified the poor student with his cold cybernetic stare.

- We’re there – Bit said when they came close to the white blocky door. A group of nervous students standing nearby was busy looking through their notes. – This ghost-room cannot be entered from the corridor. We can get there only through Pietraszko’s room.

- It is not the best moment. They are having an exam right now – Beta said.

Suddenly the door opened. The boy who left the room was beaded with sweat, tired, yet his face shined with relief. He passed his friends and headed to the stairs with an absent look.

- Zięba! – they heard from behind the door. The student froze.

- Zięba! Your student book!

The sense of relief returned to the student’s face. He entered the room, took his student book and went to the stairs again. This time he showed his curious friends three fingers.

- He passed the exam; he got C – Bit guessed.

Beta and Bit stood in silence for a moment. They were wondering what to do next.

- Artur! Beata! What are you doing here? – they suddenly heard a familiar voice. Only adults called them by their real names. Their friends always used the nicknames - Bit instead of Artur and Beta instead of Beata. They were both interested in science and they likes these nicknames as they expressed their interests. Bit specialized in electronics. He loved to program robots that he himself constructed. He also liked to break computer protection systems. Beta’s hobbyhorse was statistics and data analysis. Her nickname was connected with that interest – beta is a Greek letter frequently used in statistics.

-Has anything happened? –asked professor Jacek Błaszczyk, their dad’s friend. His academic title did not suit him at all: he had recently turned forty but you could always see him wearing jeans, unzipped hoodie and a colorful t-shirt underneath. A touristic backpack was hanging from his shoulder. He wasn’t fat nor thin; he wasn’t short or tall –his appearance was completely average. Yet his scientific achievements were not ordinary at all. Professor Błaszczyk, or just Jacek for Beta and Bit, worked on computational molecular medicine. His study was filled with screens displaying the newest results of the search for the relationships between applied therapies and genetic markers of the patients.

Bit always liked to talk to him but right now he was preoccupied with something else.

- Ehm, nothing. We were bored at home and we just decided to come– he explained lamely. Jacek’s expression showed that he wasn’t convinced.

- We already have holidays – Beta added just as if holidays was a normal reason for teenagers to come to a university.

- It’s actually good that we met - fortunately professor Błaszczyk stopped to pursue the issue. - Artur, I found the mathematical problems that I had told you about. They are in my desk. If you wait for a moment I will give them to you. And then we can go for a dinner together. I only need to call on Mr. Pietraszko.

Beta and Bit nodded their heads. The blocky white door opened again.

- Come in – professor Błaszczyk almost threw the teenagers into the room. – My dear colleague, may I take you a while?

Bit was ecstatic. He would have never expected that they would get into Pietraszko’s study so easily. They only had to identify the mysterious room. Bit started to look around searching for some door. Unfortunately, the walls were all covered by old bookshelves. He immediately lost his enthusiasm.

- It’s hopeless. There is nothing here – he thought with resignation. The only door in Pietraszko’s room were the entrance door.

- …. leave you for a second – the teenager barely heard what his dad’s friend said.

- What? - Bit asked.

- We need to leave you for a moment – professor Błaszczyk repeated. – My dear colleague, you don’t mind, do you? – he asked Pietraszko.

Pietraszko minded very much. It was enough to look at him to guess that much. He did not like anybody to enter his room and much less so leave there anybody while he was absent. Especially these two kids which he really disliked. Their curiosity annoyed him. He knew that they would poke around and read everything that they would find. They might even touch his things! But he didn’t protest. He just gave them a short false smile.

- The students will wait – he said. And they left.

- There is no door here – Bit told his sister the moment Jacek and Pietraszko left the room.

- Who said that this room has to have door? – Beta observed. – If that secret room really exists, only somebody off his nuts would enter it by door. The entrance must be concealed just like the room itself.

Bit looked at his sister with blank amazement. He did not suspect that she was a talented spy. He always perceived her as a polite girl. Too polite for such madness.

They started to look closely at the bookshelves and cupboards standing around them. They touched the doors, pulled the handles, moved various objects. It seemed to them that it was enough to press something to make the entrance to the secret room reveal itself. Just like in an adventure movie.

- It’s no good. We‘re wasting time. Jacek and Pietraszko’ll be back in a moment. There is no secret room here – Beta said. Bit’s idea seemed absurd to her from the very beginning but she was somehow disappointed – she’d already tasted the feeling of adventure.

- Or maybe we don’t know how to look for it? - Bit was staring at his tablet. Suddenly he understood. According to the plan the room they were looking for partially overlapped Mr. Pietraszko’s study. But it did not necessarily have to be located on the same floor. The plan may be misleading. – Maybe that room not only lacks door –maybe the entrance to it is not even located in one of the walls?

Bit and Beta fell to the floor immediately. They swept the old wood block floor looking for some kind of hatch. This time they were right. There was a small entrance right under the window hidden under some rummage belonging to Pietraszko. Their fingers detected a regular unevenness on the floor. They moved Pietraszko’s belongings to a different part of the room and they got down to opening the hatch. It turned out to be quite easy – they just needed to pry it open with a pointer.

They saw narrow steep stairs. Before Beta managed to say anything, Bit was already going down the stairs. The girl cautiously followed her brother. Inside they saw only pitch black darkness. The only light came from the window in the wall above the hatch. Unfortunately, the stairs soon took a turn. They had to grope their way in total darkness keeping close to the walls. They passed the second turn and suddenly they saw a pale light in front of them. They went further down until they reached a hall illuminated by several screens turned on.

- There you are – Bit said with triumph in his voice. He quickly approached one of the large LCD screen and started to look intently at the flow of numbers it displayed. After a moment he went to another screen. And then to another.

- It looks like a world management centre to me – Beta joked as she was looking at the computers standing in a semicircle. – Careful, lest you find a proof for the Pietraszko’s law. Don’t touch anything. We’ve caused enough trouble already.

It seemed to her that they were in the cellars of ELKA. Length of the stairs, two turns –it was enough to pass the ground floor and go further down.

Bit continued to examine the screens. When he checked all of the screens he started to look intently at the room they were located in. The other side of the room was even darker. Light coming from the monitors did not reach that place. Suddenly Beta heard a knock.

- What’s happened? – she asked.

- I bumped into some table. It’s too dark to search this place– Bit concluded. – I think we can leave now. Jacek and Pietraszko’ll be back here soon.

He was already heading for the door when the table he crashed into suddenly lighted up. Bit moved back immediately.

- Beta, look, there is some console with strange gauges.

Beta walked up to her brother. – Come now – she tried to rush him. Bit didn’t listen to her.

- What is that? - Bit took in his hand a gadget lying on the console. – It looks like a watch. It displays the same numbers as one of the screens on that machine. One, eight, seven, six.

- Leave that – Beta pulled at her brother’s arm.

- This device is weird. I wonder what’ll happen if I press this button.

- Bit! No...

Chapter 2

What Beta and Bit saw resembled an illustration from a textbook on visual arts or geometry. All around them there were rows of regularly placed columns. The rows were so long that the last columns were barely visible in the darkness of the hall. The columns supported semi-circular vaults similar to domes covering side chapels in churches. Each brick was clearly visible. They created arches linking the columns; they run up the vaults yet they didn’t meet in the central points of the domes. The tops of the arch vaults were open and through these holes a pale light was falling into the hall. The perspective was incredible and funny at the same time –it was a mysterious place which looked like an empty box of chocolates turned upside down. Yet it evokes different associations in Beta and Bit. They had a feeling that they were in an old basement. They could feel sand under their feet and all around them they saw discarded planks, tools and wooden scaffoldings. The earthy smell of freshly upturned ground only intensified that feeling.

- What’s that?! - Bit was completely confused. – Alternative reality game or what?

- This time you really came to the acid. I always knew that your ideas would finally... – Beta didn’t manage to say anything else. Somebody gave her back a nudge.

- Aaa wy szto, rebyata?! Otvali! Nu paszli!

Beta and Bit turned around. Behind them there was a rather short man wearing a dark jacket and a visor cap. They had seen such clothes only at the old family photos that their grandpa used to show them. The man grabbed them by their necks and pushed them forward. They quickly went along the rows of columns. The silence of the basement was broken by the cries of the man. They understood no word of them. They hoped that the man would guide them to the exit.

He liked to walk around Warsaw. The Old Town, the Saxon Garden, Krakowskie Przedmieście Street, Łazienki Park. But also Powiśle. The poor Powiśle with its poor, hungry and dirty citizens. Sense of responsibility, not only journalistic in origin, and his solidarity with the vulnerable members of the society made him look into the darkest corners of the city. Aleksander Glowacki preferred only one thing to walks through Warsaw: summer walks around Nałęczów. That health resort was his favorite place of relaxation. Now, however, he was standing not in front of a spa house but among heaps of sand. He was brought here by curiosity. Will the city, sinking in the flood of pigswill and waste, finally manage to conquer the pervasive filth? Is it possible to get rid of that pungent stench attacking at the very turnpikes of the city? Finally, will the citizens of Warsaw stop dying before they really enter adulthood thanks to the sewage system? They were being killed by lack of clean water combined with lack of knowledge of the basic hygiene rules. Private wells located right next to privies, rubbish stored at dooryards, stinking open gutters –that was the everyday landscape of one of the biggest cities of the Russian Empire. That landscape didn’t belong to the outskirts of the city –things were pretty much the same in the very centre.

The idea to build a sewage system in the city wasn’t new but its realization begun only two years before. The man who was persistent enough to overcome bureaucratic difficulties , get round Saint Petersburg and deal with protests of the citizens who did not like the changes resulting from the works performed in the area of their beloved Saxon Garden was (ironically) a Russian general, president of Warsaw, Sokrates Starynkiewicz. At the same time the designer of the project, a famous British engineer William Lindley, who had designed sewage systems for many other European cities was a guarantee that Warsaw would get one of the most advanced sewage networks. As Lindley was already an old man at that time he handed over the construction to his sons and soon later he also handed over to them his engineering company.

Głowacki was brought out of his reverie about Warsaw by a noise. He saw a shouting man leading two teenagers in front of him. Suddenly the man threw them and loosened his grip. The force of the blow was too much for the girl. She lost her balance and fell on the sand.

- What are you doing to these kids? - Bit was helping his sister to her feet when some man came to their defense. His clothes were very smart: he was wearing a jacket, waistcoat, tie and a bowler hat. Bit also noticed his moustache, small pointed beard and round glasses.

- Pay attention to your newspaper, sir – their torturer said, this time using Polish but speaking with a strange accent. - Sir... Sir.. ja nie znaju, kak wy nazywajetie... ser... Głowacki – the man could not find the words or make up his mind what language to chose. - Dlia progulki v parkie. To the Saxon Garden. Pust' tam tiemy, kotoryje wy ishchetie. Wot... Wot... sierieznyje.... Serious things are happening here – he added at the end.

- I know. That’s why I’m here. – the man in the bowler hat didn’t lose his nerve.

- Let’s get out of here – Bit whispered to his sister. He wasn’t going to overhear their discussion. It was the right moment to escape. He took Beta by her hand and they run together through the sand until they hid behind some big heap a dozen meters away.

- What’s happened? Where are we? – Bit asked not even hoping for an answer.

- Have you got that watch? - Beta was very alarmed. - Bit! – only now the boy noticed that he was still holding a gadget that he had taken from Pietraszko’s secret room. – What was written there?! –the teenager didn’t even try to speak. She snatched the strange object from her brother’s hand. Yet before she as much as looked at it they had company again.

- What do you mean where? In Warsaw – a dirty boy in filthy clothes smiled at them. He couldn’t have been more than eight years old. – How did you manage to get there? What’s inside? Is there water already?- he showered them with questions. – Well, never mind, maybe I’ll check it myself. Come on, let’s better get out of here – he said. Beta and Bit agreed with him on that thing. Bit already wanted to follow the boy when Beta hissed: - Bit! Let’s go back!

- Calm down. We’ve got nothing to lose. Anyway, we don’t really have time now to deal with that weird watch.

- Are you coming? – their new friend urged them.

Walking on the sand was difficult and it took them a longer while to get to the road. Only then Bit looked behind his back. Some way off in the distance they could see a solitary tenement house and several low shanties. Right in front of them they could see a very extensive sandy dug up area. A dessert? No, rather a huge building site. Bit noticed that there were some enormous bubbles partially hidden under the heaps of sand. It looked as if somebody had dug into the ground several gigantic balls and had left his work unfinished. Or as if somebody had turned a huge empty box of chocolates upside down and put it on the ground. The area was evidently still under construction –there were men standing on the domes rising from the ground. Most of them were wearing white shirts and they were busy with their work. Next to them there was a redbrick tower which seemed vaguely familiar to Bit.

The boy suddenly understood that he was now looking from the outside at the same construction that he had been looking at from within a dozen minutes ago when he had been held by the scruff of his neck by that stranger. The bubbles which caught his attention were the vaults supported by the columns. He was sure that they’d got to the centre of that strange building through Pietraszko’s study.

- I’m Antek – they boy they’d just met said finally. – I’m coming home. Maybe I’ll meet my brother and get something to eat from him. He works as a servant and sometimes he brings me different things from the kitchen. You can go with me if you want.

- Sure – Bit already forgot about recently experienced fear. He felt the call of adventure once again. He wanted to learn more about that place. He felt as if he had been reading a science fiction book of which he was a protagonist. – I’m Bit and this is Beta – he said quickly and he followed their new acquaintance.

Beta was dismayed. She didn’t feel like going for any walks in that place. She only wanted to go home but in the end she went after Bit with resignation. She took a quick look at Pietraszko’s gadget. – One, eight, seven, six –she repeated in her mind. – Oh, my! This must be a date. We’re in 1876 in Warsaw, just like that boy’s said. So it should be enough to set that watch at 2015 and we can go home. How to do it? What should I press?

Beta would probably soon –in her opinion – find out how to get home if not for the smell that she suddenly sensed. She looked around. She wasn’t sure how far had they gone but the landscape had surely changed. The heaps of sand had disappeared and town buildings appeared all around them. They went past wooden shanties, low brick houses and tenements. There were more and more tenement houses. At the same time the smell became more and more annoying. In fact, instead of smell it should be frankly called a stench. A dark foul slit was flowing down the ditches at both sides of the street. From time to time different citizens poured another buckets of dirty water into the open sewers.

- What a stench! I would never guess it’s Warsaw –Bit said. – And I thought that I’d never see anything worse than Pietraszko’s beard and hair – he joked.

Beta tried to look for some similarities between that place and the city she knew but none of the tenement houses looked familiar to her. The girl couldn’t even say where they were at that moment. She guessed that they must have been somewhere near the centre as the street was cobbled as opposed to other streets they’d passed earlier. And there was more traffic. There were many cabs on the streets; some of them were very elegant and showy, the others rather simple and modest –they were probably built for less affluent people. From time to time a cart drove slowly past them. There was also traffic of pedestrians on the pavements: smart men in bowler and top hats and women in long dresses were having a walk. The women were unnaturally bent as if the weight of their hats and head dresses was too heavy for them. Or maybe it was the weight of the dress supports underpinning the trains of their dresses? There was also a different group living among these noble souls –its complete opposite, the poor. These people were dirty and their clothes were ragged. Beta observed that they were often sad. The poor tried to survive in the city engaging in all sorts of activities –women were selling fruit, girls were offering flowers, boys –newspapers. There were also beggars who were mostly ill or lame.

- Hi, Staszek - Antek stopped in front of one of the beggars sitting on the ground. He might have been Beta and Bit’s peer. – You’re not a servant anymore?

- I quitted – the boy said. – I earn here the same money and I don’t have to work so much. And nobody beats me here.

Both boys smiled. Beta noticed that one of the beggars, apparently horribly crippled, just stood up and went away.

- I’m off, maybe Bolek’ll come today – the eight-year-old explained to his friend and he went away. Beta and Bit followed him.

- That Staszek.... his crippled legs... is it from beating? – she asked.

- No way! – their new friend still kept on smiling. – He’s ok. He just pretends to get some money. Like everyone else.

Beta and Bit exchanged glances. Apparently not everything changed – Beta thought.

- I know where we are – Bit was a little breathless from walking. – There, at the very beginning. That tower. I know it. I’ve just remembered. These are Water Filters. That basement with vaults...

Beta interrupted him. – Oh, yes. We went there with dad. Exactly a year ago, during holidays. A nice place to visit. Mysterious.

And she added: - I also have something to tell you. Everything indicates that Pietraszko’s gadget is a time machine. We’re in 1876 –the device displayed that number at the beginning. We only have to enter 2015 to get home. Easy, isn’t it?

- I don’t know – Bit didn’t seem convinced.

They reached a vast square. A long building with low towers at its both sides was towering over the square. The siblings stopped and Antek looked at their amazed faces with satisfaction.

- Station. That is, the railway – he explained proudly.

However, it wasn’t the station what impressed the teenagers so much. They were amazed by what was going on on the street. They saw a multitude of one and two-horse cabs. People were shouting everywhere. One glance told them that that place wasn’t governed by any rules. The cabs were overtaking each other from both sides and they were driving straight through the middle of the square. Some of them were scurrying while others stopped suddenly because the driver met a friend with whom he wanted to chat. The pedestrians –fashionable ladies arm in arm with elegant gentlemen and noisy paperboys –were walking among them. At the top of that a horsecar was approaching the square and soon it would drive in the very centre of that street gathering.

- We’re gonna get through this? - Beta was sure some of them would get hurt. She saw a moment ago how one of the paperboys fortunately avoided being trampled by a horse after one of the cabs driving from the other side knocked him down.

- It’s always like that at Marszałkowska Street– the boy said with an air of an experienced fixture.

Marszałkowska Street ... finally something familiar –Beta thought. Yet apart from the name that street had nothing in common with the street she knew. Nothing else was the same.

- It looks strange without the Palace of Culture. Completely differently, like something was missing –Bit said. The Palace was for him the central point of Warsaw, its symbol. He thought it more important than even the Mermaid of Warsaw.

- The towers of the station look like tiered cakes– Beta noticed.

Their companion didn’t wait for them to finish their conversation. He stepped on the street. Beta and Bit snapped out of their meditation and quickly followed their guide.

- Get out of the way! – they heard somebody shout next to them. They turned around in that direction and they saw a horse racing right at them. They stood petrified. – Out of the way! – the cabdriver repeated. Antek reacted immediately. He grabbed their hands and pulled them aside. The cab brushed right past them. Yet the three of them were still standing in the middle of the street. The cabs driving in both directions didn’t let them cross the street. They waited for a moment and finally they got to the other side of the street.

- Stinking drunkards – the boy murmured like an adult. – You’ve got to watch out with these drivers. Now it’ll be a bit more peaceful. – he promised. In a moment they went in between the tenement houses. Beta and Bit smelled the stink of the sewage again.

Józef Wąs tried to clean the sewers. Faint-heartedly. It was a dirty job, stinking and badly paid. He cleaned only when he had to. He cleaned the bin for dishwater standing in the middle of the yard once in several weeks. He cleaned the latrines once a month. He wasn’t afraid of any control. The oberpolicmajster didn’t come here. And anyway, nobody complained. It was even easier with the sewers –the rain washed them. Unfortunately, it hadn’t rained for some time and Józef Wąs, the caretaker, had to get down to work finally. Today he didn’t feel like working more than usually. He hadn’t slept well because he’d had to get up during the night to open the gate. These 10 kopecks he got was not a satisfactory compensation for the night interruptions. Yet when he was looking at the sewers awaiting his attention he thought that he wouldn’t mind opening the gates at all, even in the middle of the night, if that was his only duty.

When Józef Wąs was thus struggling with himself when he noticed a boy coming in his direction. I know that kid – he thought. Two teenagers were following the boy. They were wearing outlandish clothes. The caretaker had never seen them here. The ‘kid’ took an apple out of his pocket, ate it quickly and went past the caretaker.

- Just try –Józef Wąs whispered.

The boy threw the apple core to the sewer and started to run. The other boy and the girl sprinted after him. The caretaker also started to run. He tried to reach them with his broom but he missed.

- Get out of here, snotnoses! You think that’s funny?! – he shouted. He threw his broom on the ground with anger and went to his gatehouse. Józef Wąs would not work in these conditions. And then he realized: in that small poor gatehouse he wanted to return to right now lived all his family –seven people in one room. Józef Wąs didn’t even enter. At the threshold he turned around and went to the gate. He took the broom from the ground and started to sweep the yard wondering if he had made a good decision when he decided to leave his village.

Antek was running and laughing loudly. Beta’d just realized that as they walked through the city they often passed fruit stalls. Was that where their friend got his apple from? Did he steal it? They passed other gates. Two men standing in one of the gates were trying to reach some bargain.

- For few roubles you may buy here even a watch –explained their eight-year-old friend.

In front of another gate an older boy was inviting the passers-by to play a game of three cards. – Black –you lose, red –you win –he was repeating. They heard a joyful cry of a winner. Several people stopped –success encouraged them. Will they be lucky as well? Beta and Bit also stopped but Antek didn’t let them watch. He grabbed their hands and pulled energetically.

- There’s always somebody who falls for this – their friend said. Beta and Bit didn’t understand. – The winner was the cardplayer’s partner. His role is to encourage others to bet – he explained.

- This place is a jungle – Beta said. – No laws are respected here, maybe apart from this one –the law of the strongest. I wouldn’t like to live here.

They entered one of the gates and got into the backyard of a tenement house. The high walls casted dark shadows. Yet even lack of sun didn’t manage to hide the fact that the yard was flooded with garbage. What is more, it was crossed by a network of sewers full of waste. Like rivers, the stinking sewages had their origins, springs. They started next to each stairs. Above each of these ‘springs’ there was a container filled to the brim with a disgusting slurry. Beta and Bit saw a woman go out of the house and pour a bucket of swill into the fetid tub. Surprisingly, the content didn’t overflow. In addition to that there was a latrine in one corner of the yard. The stench of the latrine mixed with the smell of the sewage and literally got stuck in their noses.

- Antek! – somebody shouted and the eight-year-old turned in the direction of the gate. A boy was calling up his name and waving his hand. Antek run to him and Beta and Bit followed –very willingly. They didn’t feel like staying in that yard even for another moment.

- Here you are, bro – said the stranger and he gave the eight-year-old a lump of bread.

- This is Bolek, my brother – he introduced his brother. – What else have you got there? – he asked seeing that his brother was squeezing something under his arm.

- It’s a newspaper.

- May I see? - Bit asked but he didn’t wait for the boy to answer. He took the newspaper from him and started to leaf through it. Beta quickly came close to him. They immediately noticed the year of publication: 1885. The newspaper covered mainly topics such as hygiene, lack of order on the streets, accidents caused by coachmen. Name of one of the authors struck their eyes: Bolesław Prus.

- Give it back– Bolek demanded impolitely. He snatched the paper from Bit’s hands.- I’m off, kid. Maybe I’ll come tomorrow.

He went away. Antek followed him with his eyes. After a moment he run after him and took him by his hand.

- Beta, have you got that gadget? – Bit asked suddenly. – Switch it to 2015 and let’s go home.

- It’s not going to be that easy. I still don’t know how it works –Beta tried to convince him.

- Give it to me. Quickly, before that kid comes back – he hastened his sister.

Bit touched the display intuitively and he entered 2015. Then he pressed the button.

Chapter 3

Ruins – this word brought to Beta and Bit’s minds castles, palaces or even manor houses and churches visited during school trips. Another close association was boredom. The ruins they knew best was a castle in Kazimierz Dolny. They used to go there with their dad two or three times a year. A must of each such trip was climbing the ruins of the castle located at the top of the hill. For Bit and Beta it was just few battered walls with window openings. Bricks or stones, even if they were very old and historic, were definitely not their cup of tea.

Yet what they could see in front of themselves had nothing to do with a school trip. First they smelled dust and smoke mixed with the smell of crushed bricks. The air was so heavy that their eyes started to water. Beta sneezed. Only then they realized they were standing among ruins. In the sea of ruins. The weather was very nice and the dust was clearly visible in the rays of the sun. All around them they could see stumps of houses instead of old tenements. Their naked entrails revealed fragments of individual flats. The surface of the street was covered with rubble as well. Yet the people seemed not to notice that. There were new paths marked out in these strange mountains. In some places grass already started to grow. A boy wearing rags was carrying a bucket full of water. His feet were used to the broken bricks covering the ground –his steps were sure as if he was walking a straight path instead of the rubble.

- This time we’ll not talk to anyone –Beta quickly warned her brother. Bit just nodded his head. – We must find out where we are – she continued – and first of all –what years is it?

- Let’s get out of this mess. Maybe things will get clearer if we move from here –Bit said. Sure, but which direction should they choose? Beta and Bit had absolutely no idea. They saw a young woman going along a path covered with rubble next to them. She was carrying a little baby wrapped in a dirty blanket. She wasn’t walking, she was stomping slowly, as if she was deep in her thoughts. She was lulling her baby all the time and apparently that helped – the baby wasn’t crying.

- Let’s go after her – Beta proposed. – If she has such a small baby, she must live in a proper house, not in these ruins.

They went after the woman along the rubble. They didn’t follow her too closely because they didn’t want to scare her. As they were walking they were trying to find some familiar sign or detail which would allow them to guess where they were. They went past a group of playing girls. Their play was strange –it involved no action, just words.

- Good morning.

- Good morning.

- Is your house still standing?

- It does, but it’s burnt down. And yours?

- Mine has collapsed. I’m looking for a new flat.

What must’ve happened here – Beta wondered – that these little girls transferred that miserable reality into their games?

They reached a square. That is, it might’ve been a square in the past. Now there was only more space in here than in the surrounding area. A small group of people was looking intently at something laying on the ground. Beta and Bit came closer not losing sight of their guide at the same time. They saw a fallen statue of a king with missing hands. The people were looking into the cold eyes of the statue in silence. The brother and sister went after the women lulling the baby again. There was a little less rubble on the ground. The buildings looked more and more like original buildings and not a pile of collapsed bricks. They were burnt, black from smoke, the panes of their windows were broken, but their destroyed facades told of their former grandeur.

They passed by a destroyed palace with a colonnade. A teenage girl was selling bread spread on a low stool next to the broken columns. As she had no customers, she was sitting on the ground and reading a book. There was some park next to the square. A young boy was looking after a goat grazing on the grass. They saw another palace oriented perpendicularly to the palace with the colonnade. It was also huge, several tens of meters long and completely burnt down.

Beta was lost. Bit too. They were looking at a dead city. However, the most depressing sight were not the destroyed buildings, their empty windows or lifeless faces of the passers-by. It were the crosses which they could see particularly everywhere; crosses made of whatever was at hand: sticks, planks. Sometimes the graves were marked only by simple sticks inserted in the soil with a paper saying who had been buried in them. That city was a double cemetery –its citizens, as well as its architecture, were dead.

They walked for another several hundred meters.

- I’m not sure whether it was a good idea to follow that woman – Bit said. – She seems to be going nowhere, as if she had no place to go to, no home. Maybe she just wanted to lull the baby to sleep?

They went into another street with high four - or five - storeyed tenement houses, burnt just like others. In many cases they were completely empty inside, without walls or ceilings. The only remaining elements were the decorated facades. In some places, especially in narrower streets, the piles of rubble reached the first floor. One high building towered over the whole area. It was very narrow and rectangular in shape; in many places its regular skeleton was clearly visible making the building resemble modern blocks of flats. Now they knew where they were. It was Warsaw, for sure. That high building still stands at Świętokrzyska Street. Grandpa called it Prudential, just like his mother did, but father called it Hotel Warszawa. But neither Beta nor Bit remembered if the building ever housed anything of consequence. For them it had always been empty, closed and dilapidating.

-Let’s leave that woman and go there – Beta pointed the way ahead. – We’ll get to Marszałkowska Street.

They set off again in a quicker pace to get closer to their guide who was still lulling the baby. As Beta was going past her she looked back to see the child. To her surprise she saw a baby doll wrapped in a blanket. The woman was lulling a doll all the time. Beta was dumbfounded. After a moment her horror changed into sadness, then anger and finally into a strong wish to escape from that wounded world. The girl started to run through the rubble in the direction of Marszakowska, as she though. She stumbled several times and finally she fell down. Bit run after her and helped her to get up.

- Easy – he tried to speak calmly and slowly. They sat still waiting for Beta to regain her tranquility. Bit looked into a nearby street. He saw several boys playing at war, pretending to shoot at each other from guns made of sticks. –Let’s go. At Marszałkowska Street we’ll think what to do next –Bit decided finally.

Despite the ruins they got to Marszałkowska Street quite fast. Once again they were surprised not to find there the inseparable symbol of Warsaw, the Palace of Culture. They saw another high building on their right. They recognized it although a significant part of it was destroyed. They used to go there with their grandpa. Grandpa called it PASTa. Beta and Bit didn’t remember what that acronym stood for exactly. They only knew that the building housed a telephone exchange before the war.

The brother and sister eventually decided to turn left in the direction of Jerozolimskie Avenue. There were more people. Some woman was hanging laundry on a string running across an open storey in one of the destroyed tenements. Streets were occupied by carts and two-wheeled handcarts with people’s belongings, or, more specifically, with what was left of them. There was also a funeral procession attended by several mourners following the cart with a coffin. They heard a distinct sound of the clatter of horses’ hoofs. A woman was sitting on the pavement with her back against the wall of one of the tenements. She was holding in her arms a sleeping little girl, her daughter. Next to them numerous sellers were plying their trade. An older woman was selling flowers, some other person –bread. There were new signboards above the ground-floor apartments of the destroyed tenement houses: chemist’s, hairdresser. Two ten-year-old boys were walking before Beta and Bit. One of them was walking on crutches –he had lost his right leg. In his free hand he was holding a ...

Suddenly the whole world started to tremble. Its edges became blurred; everything became strangely faded. No, it wasn’t the world; it was just what they were seeing. The sound of the street died away. Instead Beta and Bit started to hear a sound resembling a wailing wind. It seemed to be coming from some chimney. At first it was quiet but its volume rose with every second until it changed into...

- I’ll try to get them back– Mr. Pietraszko, MSc was standing next to the console with a determination visible on his face. When he’d returned to his study with professor Błaszczyk the teenagers were nowhere to be found. Pietraszko didn’t care at all. On the contrary, he was happy that these annoying kids got out of his face. His companion wasn’t preoccupied either.

- They came up with something again and went somewhere. Well, I’ll have to eat my dinner alone after all- he said. – Or maybe you would like to come with me? – he asked unexpectedly.

But Mr. Pietraszko didn’t hear that question. He noticed that the trapdoor in the floor under the window had been lifted. He went mad. So many years of work, sacrifices and scorn of his colleagues. So much effort to keep the work of his life secret. And it’s all for nothing because of these two meddling kids. He jumped to the trapdoor completely ignoring his guest. He run down to the very bottom of the stairs. Then he rushed to the screens in order to try to read something from the displayed data. Then he jumped to the console placed at the other end of the room and he started to search...

- It’s gone –Pietraszko said and his anger started to give way to terror. - My remote is gone.

- What’s this? What’s that place? - professor Błaszczyk asked amazed. He followed his colleague down the stairs. Pietraszko turned back. He looked possessed. His face twisted in anger was only partially covered with his long hair.

- I’ve been working on this for many years – Mr. Pietraszko muttered through clenched teeth. – Against the opinion of the scientific community, even against the common sense. And I succeeded.

- Excuse me? – professor didn’t know what Pietraszko was talking about.

- And these stupid kids destroyed everything! – he shouted.

- What? Where is Beata and Artur? – asked professor Błaszczyk with raising alarm. He felt responsible for them no less than their parents did.

- Who knows? – Pietraszko said maliciously. – Their curiosity became their undoing.

- What are you talking about? Do something! - professor was speaking with an increasingly raised voice.

- It’s not that simple. The machine is not working properly yet. There are errors– Pietraszko drew in his horns. He didn’t like when people shouted at him. He felt uneasy.

- What’s going on here? Kindly explain it to me! –professor Błaszczyk cried.

- Later –Pietraszko said quickly. –I’ll first try to trace them and bring them back.

The roar of sirens was piercing their heads. A long loud shrilling and high sound, then a short and low sound. And then again: long high shrilling sound and a short respite. And again: long... At the same time they could hear powerful rumbles which made their ears ache. The ground was trembling. Yes, this time for sure it was the earth. Fire, smoke, smell of burning. The world was becoming less and less blurred. The tower of the Royal Castle was on fire.

- What’ve you switched? – Beta shouted through the roar of falling bombs.

- Nothing, this time it wasn’t ...

Bit didn’t finish. The world faded again and the sounds became more and more distant.

- It didn’t work – Pietraszko was strongly focused. – I’ll try again. There is some error – he started to explain and then he stopped. He stared at the data from the screens again.- If I fail again they’ll have to calculate the way home by themselves. It’s actually good they like science; it may be their only hope for returning home. – Mr. Pietraszko could not hold back his malice.

This time there was no roar, no noise, no blurred world. Beta and Bit suddenly found themselves in a line. People were standing one after another; it was very hot. Somebody pushed some heavy object into Bit’s hands. A brick! The boy couldn’t believe that.

- Pass it! – his neighbor shouted. It was the same man who gave him the brick.

Bit passed the brick to Beta and at the same moment he had to turn back to receive another one. It was heavy and it scratched his hands. Next bricks had even sharper edges. Bit was confused. He hadn’t pressed, hadn’t switched anything on that weird watch. Why were they transported again? Why were they thrown into different years like that? He wished he could talk to Beta but the man next to him passed to him yet another brick. Bit didn’t take it. The world became blurred again.

- It’s not going to work– Mr. Pietraszko stated categorically. – We can’t do anything from here. These kids took the main control element. They have to find the way back by themselves wherever they are and whatever past sucked them in. I can’t help them.

Professor Błaszczyk didn’t quite understand what his colleague was talking about. He only guessed from the expression on Pietraszko’s face that the situation was very serious.

High buildings loomed up in front of them. They could see them more and more clearly. They were destroyed and burnt tenement houses, black from smoke and without windows. Beta and Bit were back on Jerozolimske Avenue.

- I haven’t switched anything –Bit started to explain immediately.

- I think something is wrong. It’s getting dangerous. It’s going to throw us in the middle of some war again and we may be less lucky than we were a moment ago– Beta was really scared. – We must find a way to get back to our times.

Bit hadn’t got the slightest idea how to do that. – I switched the machine to 2015 and we got God knows where. I’ve got no idea why. And in addition the machine transported us twice of its own accord. We succeeded only the first time.

Beta mused for a moment. – Not necessarily – she said. – When we were first looking at the machine in Pietraszko’s study it was switched to 1876 but, as far as I remember, the newspaper that you took from that boy, Bolek, was from 1885.

- The machine makes mistakes –Bit concluded.

- If these differences are random, the possible variants are innumerable and we stand virtually no chances of getting home – Beta observed.

- So let’s assume that the shifts are not random and that there is some relationship between them. It’s not very realistic but we’ve got nothing to lose.

It was hard not to agree with him.

- All right. Let’s treat this like a mathematical puzzle. – Beta loved to solve such problems. – We need to gather all the dates: the dates from the watch and the years we got into.

Beta’s confidence rocketed. She was in her element.

- It shouldn’t be difficult, there were only four dates –Bit remarked with a smirk. – The first pair is 1876 and 1885. The second pair is 2015 and the year we’re in currently. We only don’t know what year it is now.

- Let’s think... - Beta hid her face in her hands to concentrate better. – It must be after war, after the II World War. I can see no German soldiers or hear any bombs or shots.

- But the war must’ve ended not that long ago either –Bit joined her. – There is too much rubble, too many ruins and street graves. Nobody is reconstructing the buildings.

- These carts with bundles... I think that the people are just returning to their houses after the war. Let’s assume that it’s 1945, a year after the end of the II World War. – Beta was very pleased with herself. – I wouldn’t take into account these two places we were transported into by the machine itself. We didn’t set the machine then.

- Even if you’re right about the year we still have got a problem – Bit seemed determined to spoil his sister’s pleasure. – Two points are not enough to detect any relationship. Whether you want it or not, we need to jump in time once again.

- Are you mad? Haven’t you seen what’s happened? - Beta was unsettled again. – How can you know where we’d be transported?

- Two points aren’t enough –Bit repeated. – We simply have to risk. The higher number I set, the later year we get into. I’ll just write a much higher number this time –the teenager pressed four numbers on Pietraszko’s gadget without further ado: 2098.

- Bit! No... – Beta’s cry stopped.

Chapter 4

- Dear children! What are you doing here? - Beta and Bit saw an older man standing in front of them. It was very cold, icy cold. Bit knew that feeling from books: they were cold during the teleportation; sometimes they felt pain; it seemed to them that some void was sucking them in. But this wasn’t like that. Bit was cold all the time, not only for a moment when he was travelling in time. There was no void either. The world was absolutely real and his body was suffering real freezing cold. Bit looked around. – We must’ve got back in time – he muttered looking at the soldiers standing close to a tank.

- My dears! Get inside or you will end up with pneumonia or something even worse – the older man warned them. He was wearing a sheepskin coat with a fur collar. Beta and Bit were dressed in the light summer clothes they’d been wearing when they entered Pietraszko’s study.

Two soldiers were marching in step in their direction. The man quickly put his arms around the teenagers and he pushed them lightly in the direction of the gate leading to a tenement house they were standing next to. - Citizen! – they heard one of the soldiers cry after them. The older man turned around. – They’re my grandchildren – he explained. – They went out with me.

- Take better care of them then – the soldier grunted.

The older man only nodded his head and hugging the teenagers again he led them out of that place. The soldiers turned back. – Come, I’ll make you a hot tea. You’ll warm up a bit – he invited them.

His flat was gloomy grayish and dark. Its big windows didn’t let much light in.

In the living room there was a small sofa covered with a brown coverlet, a round table with four chairs and a little cupboard on legs. It caught Beta’s attention. She had seen a similar cupboard at her grandpa’s place, only it was bigger. This one was a little higher than Bit and just like the grandpa’s piece it had a lockable drink cabinet. On the left-hand side of the drink cabinet there was a cupboard with glass door. There was also a narrow shelf shielded by similar glass door hanging above both cupboards.

- Sit down on the sofa bed – said the host. – Go ahead – he encouraged them pointing to the sofa. – I’ll put the kettle on. And you, child– the older man turned to Beta who was still staring at the cupboard – can you take out the glasses?

The cupboard with glass door was filled with small and big glasses, metal cases, brown cups with painted flowers and one white cup with rose pattern. It differed considerably from other cups. It was fragile and dignified. Beta thought that this was the only real china cup in the service. The upper shelf was a place for some crystal bowls, jugs and an ashtray. The girl took out the glasses in yellow apples – she liked these best. She put them on the table and sat next to her brother.

- Ok, so this is a sofa bed, not a sofa –Bit said.

Their host returned with a little red teakettle in his hands. He threw a glance at the table and he smiled benevolently. – My grandchildren, the real ones, also like these glasses best. Children are partial to them, it seems. Unfortunately, they cannot be used for tea – he said and he took from the cupboard three other glasses and three metal cases. Beta stood up to put the glasses into cases and the older man poured tea into them. „That’s it?” - Beta thought when she saw that the tea barely covered the bottom of the glass. The older man disappeared and after a while he returned with another kettle. – And now we add boiling water – he said to himself. – After my wife died I had to learn everything, even how to make tea – he started to explain.- This teakettle was her idea. She used to put dry tea inside the kettle filling it almost to the lid and then she poured some boiling water. She poured a little of that essence into glasses and filled them to the top with boiling water. I do it now in the same way.

He took a silver sugar bowl from the cupboard. –Oh, yes, and the spoons – he remembered and he went to the kitchen again.

- You have to forgive me, I don’t have visitors often – he said as he came back to the living room. Beta and Bit were familiar with such monologues. Their grandpa behaved in the same way during their visits. He was talking all the time; he wanted to discuss everything. – Have you got any names? – the host asked jokingly.

- Artur.

- Beata.

- Great. I’m Jan – he introduced himself. – You must be still cold, I bet. There are blankets in my son’s room. I’ll bring them in a second– the older man left for a moment.

- We’re in the period of war again – Bit said. He knew they had little time to talk. – We must’ve moved back in time.

- I think you’re wrong. Look around – Beta pointed to a small table standing in the corner under the window. – There’s a radio. And besides, the soldier was speaking Polish.

- True. But what he said was weird. ‘Citizen’ - who’s talking like that? – Bit asked.

- I’m back – their host came back suddenly. – It took a while but I found two blankets. Here you are – he said as he gave the blankets to the teenagers. – Wrap yourselves and sit at the table.

Beta and Bit obediently sat on the chairs and picked up the cups by their metal handles. They were really cold. They first froze outside and then inside the flat of their host which was rather cold as well. The teenagers were slowly sipping at their tea. It tasted like paper but they didn’t mind that. It made them warm and that was enough.

- Next time you want to get outside –the older man started – put on some coats. Although it would be better if you don’t go outside at all. At least not without your parents.

Their host took a sip of tea.

- What’s been going on these two days... – the man stopped and lowered his head sadly. – Civil war. Brothers against brothers. Poles against Poles.

Another sip of tea and silence again. Beta started to look around the room. There were pictures on the wall: a smiling woman, a young couple, a little boy sitting grandly on the armchair. And a very characteristic picture of the Mother of God with a silver moon underneath.

Suddenly they heard a sound of a key in a lock. A young man jumped into the room. He couldn’t have been more than thirty years old although his moustache suggested otherwise.

- Hi, dad! – he said. – You’ve got visitors?

- Neighbors’ kids – the host explained.

- I haven’t seen them before – the man with mustache was surprised. - Anyway. I came to take a few things –he said and he went to the other room.

- That’s my son, Krzysztof –the older man said to Beta and Bit.

- Bye, then! I’m off. Take care.

- Do you... –Mr Jan didn’t finish. The man slammed the door and locked them. Krzysztof disappeared as suddenly as he came. - ... also - Beta and Bit barely heard that word.

- My grandchildren are much younger than you – he said finally after a longer pause. – They are little children actually. Peter is five and Anna’s three. They are the best part of my son’s life. Mine too, in fact. Unfortunately, he believes that he is a member of the next generation of Polish insurgents. He wants to fight for a better Poland and he jeopardizes the safety of his family for some vague ideals and fancied purposes. And he is as stubborn as a donkey; he doesn’t listen to his father’s arguments and advice. He should listen because I know what I’m saying. I have seen occupation and insurrection; I had to run away from my native city; I was cast out of the destroyed capital and I returned to the ruins.

So we’re still in Warsaw. These are different times but we’re still in the capital – Beta thought.

The host stopped. They sank into silence. Beta and Bit were still silent.

- I’ve been afraid for these two days again –the man continued. –I’m scared for my family and for myself.

He stopped and after a moment he continued but this time his voice was completely different. – Come on – Mr. Jan stood up from his chair and went to the window. – I’ll show you something.

Beta and Bit rose up. Still wrapped up in the blankets they went up to a dirty window to look at equally dirty world. The cloudy sky was grey like the filthy snow covering everything all around. They saw the same tank they’d seen before and several soldiers.

- The tank is still there –Beta said.

- Yes, it is. Only it’s not a tank, it’s a transporter. But look where it is standing. What a meaningful coincidence.

Beta and Bit only then noticed that the tank was parked in front of a high building. At the top of the building hanged a huge word ‘Moscow’. On the building below there was also another word ‘cinema’. Lower still, above the very entrance, there was a big white poster with the title ‘Czas Apokalipsy’ (Apocalypse Now).

- Sometimes I stand by that window and look at all of that and I wonder how is that possible that none of them hasn’t noticed it yet. And then I smile to myself because it means that they are not as omnipotent as they’d like to think. Not everything depends on them. Would you like to drink some more tea? – the man asked suddenly.

- Yes, I’d love to– Bit looked as if he’d just woken up. – My sister would like more tea too –he added although Beta shook her head. The older man went to the kitchen.

- What are you doing? – she asked nervously.

- What do you mean? I’ve sent that man to the kitchen so we could discuss how to get home –the boy explained. –Let us gather the dates again.

Bit took a pencil and a newspaper he’d earlier noticed on the table with the radio and he sat by the big table.

- 1876 and 1885 are the first pair – Beta said as she sat next to her brother. - 1876 was on Pietraszko’s gadget and 1885 was the year of publication of the newspaper that we read then.

- Then I entered 2015 and we were thrown into 1945 as you claimed – Bit added.

- I still think it was 1945; I don’t have any better idea. The city looked as if the war’d just ended. And later on I don’t know where we were because you entered some year yourself, just for kicks.

- Not for kicks, I wanted to make it easier for us to get home. I wrote 2098. We only need to find out where we are now and calculate somehow what date we have to enter to get back to Pietraszko’s shelter. Maybe we’ll just ask our host what’s the year now?

- No way –Beta answered quickly. – What if he shows us the door? Do you want to calculate that out in the cold? I think that we are not very far from our times. The soldier said ‘citizen’ just like in the comedies dad likes to watch.

- The ones only he finds funny –Bit reminded.

-They are not funny for you because you don’t understand them; you don’t know the context. But dad is laughing when he watches them because they show the times when he was a child. I think that we’re in that period now. 1970s or 1980s, I don’t know exactly.

- Wow, you really limited our room for maneuvers – Bit couldn’t leave that without remark

- I’m back – said their host. He was holding a red teakettle in one hand and a kettle with boiling water in the other. –Some puzzle? –he asked looking at their notes in the newspaper which Bit inefficiently tried to hide from him. – I like puzzles, especially the mathematical ones. I used to work in the CSO for years – their host went to the kitchen and he quickly returned –this time without the teakettle or the kettle.

- CSO, that is, the Central Statistical Office – he explained – I’ve cut my teeth on number crunching. Show me what you’ve got –he said standing next to Bit.

According to Beta everything was going in the wrong direction. She was sure that in a moment they would return to the cold street and pressed by the circumstances once again they would enter some random year in that strange watch.

Bit, on the other hand, loved unexpected situations. He coped with them much better than his sister. He quickly showed the notes to their host.

- We’re trying to find a relationship between several points –he started to explain. – They are dates. We have three pairs: 1876 – 1885, 2015 – 1945 and 2098 and the current year.

- Then write 1981. What are you waiting for? – Mr. Jan said.

Beta didn’t expect that it would be so simple. Bit grinned widely and wrote down 1981.

-We also have the second element of the fourth pair: 2015. We want to find its first element – he continued.

- Have you tried to draw it? – the man asked. – From your expressions I can tell that you haven’t – he said kind-heartedly. –Let us start then. Boy, draw the coordinates. – He said taking a notebook, ruler and a pencil out of his desk. He found a blank page in the notebook and gave it to Bit.

Bit competently drew the three points describing their adventures.

- Look, these points are almost ideally placed along a straight line. You can use a ruler to draw a straight line running through these points and then you may check where that line crosses the line corresponding to 2015 – Mr. Jan said drawing a line going straight through all the points with an expert hand. – Look, you can find an approximate solution to this puzzle with nothing more than a ruler and pencil.

- But can we find an exact, not approximate result?– Beta asked. She preferred to avoid approximate results and time travels which would result from them.

- Of course, you only need to draw the best fit line of the linear regression, that is, the function with equation y = a x + b. The value of y and x are the coordinates of your points, and the values of a and b are the unknown values that need to be calculated.

- But how to it? – Beta urged him.

- This is really an interesting problem. Generally, you have to minimize the distance between the points and the line. You can do it in many ways– Mr. Jan started a lecture. – The very name of the problem – regression- comes from the expression ,,regression toward the mean’’, used by Sir Francis Galton, when he started to examine relationship between height of parents and their sons. He noticed that the height of the sons is closer to the average height of the whole population than the height of their parents...

- But how to solve this particular problem? – Beta burst out although in any other situation she would listen to a lecture on data analysis with interest. This time, however, she was frozen to the marrow, tired, scared and she wanted to go home finally.

- In this particular case – Mr. Jan’s regret was almost audible as he continued – notice that when the values from the horizontal axis changed from 1876 to 2098, that is, they changed by 222 units, the values of the vertical axis changed from 1885 to 1981, that is, by 96 units. If we divide it – Mr. Jan divided three- digit numbers with ease - 96 / 222 is around 0.43, then a one-unit move on the horizontal axis gives us a 0.43-unit increase on the vertical axis. You want to move vertically from 1981 to 2015, that is, by 34 units. By how many units do you think you have to move horizontally?

- Ooooo –Bit started a little dimwittedly. I could do with a laptop or at least a calculator.

- That would be 34 divided by 0.43 so almost 340 divided by 4, which is 170 by 2, so around 85… - Beta calculated out loud.

- And even more around it will be 79 –Mr. Jan smiled. Mental calculations were much easier for him.

- So we need to move on the horizontal axis from 2098 to 2098 plus 79, that is, 2177 – Bit wanted to obliterate the memory of his not too smart reaction.

- Exactly, that is the solution to your puzzle! See how easy it was? – Mr. Jan looked very pleased. – Now you know what linear regression is. Maybe you’ll make use of that knowledge some day.

Beta and Bit smiled widely. They’ll be able to go home finally. They chose not to think for the time being about Pietraszko’s reaction to their intrusion and discovery of his secret cavern. Was it possible that he didn’t notice anything? Maybe he hasn’t even returned with Jacek to his study yet? No, that would be too good to be true although they stood a ghost of chance.

- I haven’t had such an interesting day for some time. I forgot about the martial law for a moment. I’m really happy that I could help you. Although maybe it was you who helped me – Mr. Jan said.

- Without you we wouldn’t be able to get home –Beta said suddenly.

- That’s an exaggeration. I just lied a bit to the soldiers, that’s it – their host smiled. – You haven’t drunk your tea because of all that mathematics – he observed. – Now it’s completely cold.

- We don’t mind – the siblings said almost simultaneously. They drank their tea down in one draught. It tasted like paper even when cold.

- So maybe we’ll just go now – Bit said as he got up from the table. – Thank you for tea and your help – he thanked heading for the door. Beta followed him.

Mr. Jan opened the door for them. – Good buy – they said to their host. When he closed the door Bit grabbed his sister by her hand and pushed her up the stair to the next floor.

- So let’s check it – he said entering 2177 in the watch.

Chapter 5

Nothing more than a dull light of the computer screens illuminated the gloomy study belonging to Mr Pietraszko, MSc. Nothing happened for a long time. How long? None of them was aware of the passage of time. An hour, maybe more. They waited in silence. Yet none of them knew what they were waiting for. Pietraszko checked the communicates on the displays every few minutes trying to catch some clue, some hint as to what had happened and what to do to make that story end happily. Professor Błaszczyk had never felt so helpless before. He didn’t know how to help the teenagers. He had to rely on Mr. Pietraszko, MSc, who was surreptitiously derided by the student and openly laughed at by many fellow scientists, and on his scientific ideas balancing on the verge of sanity. Even if Pietraszko’s theories about the nature of time and events were correct, he didn’t care for it at all at that moment. He was focused on Beata and Artur.

- Something is going on – Pietraszko burst on his thought. –Markov chains no longer converge, there is no ergodicity. It looks as if the algorithm comes to some serious obstacle. In fact it’s logical –Pietraszko was talking to himself. – But they cannot jump ahead, into the future, further than 2015. Oh, well, now I see, they’re going to return to the stationary distribution – Mr. Pietraszko, MSc was very glad.

- Beata! Artur! Finally –professor Błaszczyk cried with relief seeing two people emerging behind the console located in the dark corner of the studio.

Pietraszko stopped his calculations.

- You caused a pretty mess – he said raising his eyes from the keyboard. He faced the teenagers, crossed his arms and waited. This was not a particularly warm welcome and it didn’t bode well. Bit thought that he actually preferred 19th-century tenements to this. Nothing could be worse than gloomy bearded face of unpredictable Mr. Pietraszko, MSc.

- We... - Bit said uncertainly– … didn’t want. That is... we only wanted to check where was the room we found on the map.

You found – Beta thought but she decided not to open her mouth. She let Bit explain himself.

- What map? –Pietraszko hissed and the teenagers understood that things were really bad. – Did you rummage through my private folders on my private account?

Pietraszko was turning intensively red and professor Błaszczyk could not believe what he had just heard. He knew that Bit liked to break security measures and that his satisfaction from cracking was greater than his fear of potential consequences. He knew how that felt himself. Yet he believed that Artur was wise enough not to hack the data at his father’s work. And if not Artur, then certainly Beta, who usually brought her brother down to earth.

- You broke into my own account, then to my study and then you stole my invention… – he hissed.

- Stole? – Bit was surprised.

- Yes, you stole it –Pietraszko confirmed with diabolical satisfaction. – And yes, give it back to me.

- No... I only touched it. I wanted to check and I accidentally pressed .. – Bit tried to explain incoherently as he put the invention on the console.

- So you wanted or you pressed it by chance? – Pietraszko insisted.

- You let us into your study yourself –Bit retorted.

- Let’s finish this –professor Błaszczyk cut in suddenly. – My dear friend – he turned to Mr. Pietraszko, MSc – please tell them what you expect from them.

- I’ll be honest –Pietraszko begun- and I don’t think you’ll be surprised by what I’m going to say: these kids always got on my nerves. And it’s not going to change. But I promise to forget about everything if they tell how on earth they managed to return to our times– the scientist shifted his focus from the break-in to something much more interesting to him. – The driver is not finished and it contains errors. How did you know how to calibrate it?

- Well, we had some problems with it too– Bit admitted. – Mr. Jan helped us.

- What Mr. Jan? - professor Błaszczyk asked.

- Linear regression– Beta decided that it was a good moment to join the conversation and boast of her knowledge at the same time.

- We gathered data describing the working of the watch and then we calculated the year that should be entered –Bit cut in confidently as if discovering workings of time vehicles was his everyday occupation.

- Linear regression –Pietraszko repeated with surprise. – Such a banal solution that it seems almost impossible – she said. Where did you get the data to determine it?

- We travelled in time a little –Bit answered, evidently pleased with himself.

- Even during our first travel we were thrown into a different year than we had chosen - Beta explained. – We should’ve got into 1876 but we were thrown into 1885. We didn’t care for it then. But later we entered 2015 and we got into 1945. It seemed obvious then that the machine makes errors. And in the meantime we were twice thrown into some other years for short moments.

- It must’ve happened when I was trying to find out where you were – Pietraszko observed.

- I decided than we should jump in time once more to check what was really wrong with your invention – Bit joined the story. – I entered 2098 and we got into 1981.

- How did you know what was the current year? The machines gives wrong dates – Mr. Pietraszko, MSc wanted to know every detail.

- Actually, we had a guess only once, with 1945 – Bit related. –We used a newspaper the first time –we read the year of publication –and the second time we were helped by Mr. Jan.

-You talk about that Mr. Jan again. Who’s he? - professor Błaszczyk could not wait.

- Mr. Jan protected us from the soldiers… - Bit didn’t finish.

- And helped us to solve that puzzle – Beta didn’t want her brother to rob her of the whole story. – We had four points in time: three from our jump –we knew then what was entered in the watch and what year we got into, and one point which we wanted to reach. We only needed to know what year we should enter to get back to 2015. Mr. Jan told us then about the linear regression.

- How is it possible to travel in time anyway? – professor Błaszczyk was starting to think logically again.

- It isn’t possible and I will not tell you the secret, no way –Pietraszko promised. – It’s enough for you to know that I based my calculations on the law which all of you derided. And now what? You ask me how is that possible and I’m supposed to tell you everything? None of you’ll understand or appreciate it. But the linear regression... – Mr. Pietraszko, MSc changed his tone and his face became somber. – The type of error which could not be discovered in the study. I’d have to travel in time myself to find it out – he said bitterly. He lacked the courage to do something so crazy. He’d decided it was too dangerous. And now these kids robbed him of all satisfaction. The joy he initially felt evaporated. Now he felt only regret and anger. – You’re still here? – he asked sharply after a moment. – Het out of here, I need to clean up! – he started to shout.

- So? Shall we go for these mathematical problems? And then for dinner? - professor Błaszczyk asked the teenagers. He pretended that he hadn’t heard Pietraszko’s rude question. Beta and Bit only nodded their heads; it was better to say nothing.

Beta and Bit shouted over one another relating their adventures to Jacek. They didn’t mind that they were walking along the university’s hall.

- … and that man, he was called Głowacki –Bit said.

- He protected us against some other man who was speaking Russian, I think. He said that that Głowacki was from the newspaper –Beta added.

- Where the Filters are located. Only they were under construction then. I recognized them thanks to the tower.

- Wait, you saw the construction of the filters? Głowacki? From the newspaper? - professor Błaszczyk started to laugh. – It seems that you met Bolesław Prus.

- No, Bolesław Prus signed an article in the newspaper in which we found the date –Bit explained.

Jacek quickly took out his mobile phone and started to look for something. – Did your Głowacki look like that? – he showed them a picture of a man in round glasses and sharp beard.

- Głowacki... Of course! Aleksander Głowacki was Bolesław Prus! – Beta cried.

- I think that both of you must buckle down to Polish classes–Jacek joked. – Talking about classes, I just remembered something. I left my students’ courseworks in Mr. Pietraszko’s study.

- Don’t tell me we need to go back to his cavern –Beta made a wry face. She didn’t feel like meeting the eccentric scientist again.

- I’ll go –Bit offered happily. He was afraid of unpredictable Pietraszko but his curiosity exceeded his fear.

The door to the study was still open. It means that Mr. Pietraszko, MSc hadn’t left the university yet. Professor Błaszczyk knocked but nobody answered. He entered the room. Pietraszko wasn’t inside so the professor didn’t hesitate to open the hatch under the window and step inside. The children followed him.

- My dear friend – he cried as was going downstairs.

Nothing’d changed. The computers were still working. – Haven’t I left here my students courseworks? – he asked and looked around. – Mr. Pietraszko – he cried again.

- He set off for a time travel –Bit joked.

Professor Błaszczyk went to the console. His students’ courseworks were lying there untouched. He took them and headed for the stairs when something stopped him. Not only Mr. Pietraszko, MSc was gone. The device that Bit put on the console was gone as well. Instead they saw a sheet of paper with hastily scrawled dates given by Beta and Bit. One date from a shortened version of a system of equations was circled.

- 1975 – professor read out loud.

Epilog

Beta and Bit returned to their times only because they discovered the rules governing the time vehicle. The noticed in what way the year into which they travelled depended on the number entered into the driver of the time machine and used that information in practice. Ability to notice such relationships may come in handy not only during intrusion into study of a crazy mathematician. The world that surround us is bristling with many relationships. We may use a skill of noticing and employing them in various situations.

This skill allowed the scientists to explain the Space Shuttle Challenger disaster which occurred in 1986. 45 seconds after the launch of the rocker the spectators could see a flame of fire which quickly led to disintegration of the whole shuttle. The disaster claimed the lives of the whole crew. It raised questions about the future of NASA. A special commission, which was established with a view of explaining the cause of the disaster, looked for any clues for a long time. Richard Feynman, physicist and later Nobel Prize winner, was a member of the commission. He noticed that low temperature decreases elasticity of O-rings sealing the rocket booster. Low elasticity translated into higher failure frequency. Even at temperature below 18C problems with O-rings occurred and the lower the temperature, the frequency of problems rocketed. The elasticity of the rubber rings was too small at 10C while during the night prior to the shuttle launch the temperature fell below 0C.

The following dot chart presents the results of the initial measurement of the number of temperature-related damage. All measurements were collected at temperature between 10 and 25C. It is obvious that the lower the temperature, the higher rate of damage. Had the scientist known that relationship, would they have been able to predict such a serious damage at 0C?

Regression is a method for identifying relationships between variables. Where does its name come from? Actually, the story behind it is very interesting. In 1886 sir Francis Galton was studying relationship between height of adult sons and height of their parents. He observed that sons of tall parents were usually tall but not as tall as the parents. Obviously, some sons were exceptionally tall and outgrew their parents. Yet when he examined the averages, he found out that the average height of children was somewhere in between the height of the parents and the average height of all children. Galton called that tendency ‘regression toward mediocrity’.

The following diagram illustrates Galton’s results. He examined over 1000 families. Dots located at OX axis display the weighted average height of parents while dots at OY axis present the height of children. The solid line represents the relationship corresponding to equal height of parents and children. The dashed line is the relationship reflected by the data. The average height of children is closer to the horizontal line. In contrast to the pairs of numbers collected by Beta and Bit, the measurement points are not placed ideally along the line. The line describing the relationship determined with that method is a line located as closely as possible to all the points.

Someone might ask about the value of such studies. What can we gain from the knowledge that average height of children is partially determined by the height of parents and partially it is not? Yet it turns out that the technique of linear regression paved the way for a huge progress in agriculture and cow breeding. When we know what part of cow’s milkiness is a genetic heritage coming from the bull and what part comes from the cow, we can use that knowledge to assess the breeding value of an animal. Nowadays the assessments of the breeding value are conducted on a large scale. They frequently include a complex pedigree of animals. Such assessments helped to increase milkiness of cows – it is estimated that in developed countries milkiness was tripled over the last 50 years! Such studies are based on nothing else but regression.

Now you will probably ask what is the use of regression in every-day life? Well, it is very useful and this fact can be easily proven. Let’s assume that we want to buy or sell a second-hand car or a flat or a mobile phone. We are wondering what may be the cost of a 5-year-old Volkswagen Passat with average equipment. Portals offering second-hand cars will display hundreds of sale offers. We can download that data and search for a relationship between a car’s age and its price. What would we discover? If we plot price logarithm against the cars’ age, a beautiful linear relationship will appear to our eyes. Where does this logarithm come from? It turns out that value of the cars drops with every year by a fixed percentage, which is around 15% (the exact value depends on the make and equipment). After a year a car’s price falls by 15%, after two years it falls by 27,8% (how come such percentage? here compound interest comes in handy), after three years by 38,6% and so on. However, when we take the logarithm of both sides of the price, compound interest changes into a linear relationship allowing us to quickly estimate the prices of second-hand cars.

The following diagram presents the relation between price expressed in exponential scale displayed on OY axis and car’s year of manufacture displayed on OX axis. The data was downloaded from otomoto.pl portal.

However, not every relationship is linear. Continuing the subject of second-hand cars, let us look at a very interesting relation between mileage and year of manufacture.

We would expect that mileage should increase in proportion to age of cars. If a certain number of kilometers was covered in a given year, then during two years the double number of kilometers should be covered, right? Interestingly, when we examine a chart presenting relation between mileage and age of cars, we observe a different trend for cars under 5 and a different trend for older vehicles. Young cars increase their mileage with rate of 20-25 thousand kilometers a year but after 5 years their mileage starts to increase by only 10-15 thousand kilometers a year. This dependence may result from difference in behavior of owners of older cars (sales representatives cover the highest number of kilometers and it is this group which most often drives new cars which are later on sold to new owners) or difference in behavior of sellers (they reduce mileage of the cars with high mileage and in case of older cars no service booklet is kept which would stand as a proof of manipulation).

We need to be careful when we search for relationships between variables. The fact that two variables seem dependent does not necessarily mean that one is a cause of the other. For example, when we analyze data about birth rate and number of stork nests in different villages in Poland, it turns out that there is a very strong dependence between these two values. The more nests, the more children! However, this does not prove that storks bring babies. So what does this prove? The bigger the village, the more houses, roofs, poles and other place for nests it has. On the other hand, more houses and people mean more children, obviously. This is why the number of children and the number of storks are related – they depend on the number of habitations.

Table of Contents

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Epilog