Two policemen are chasing a thief, Mathina and Leo join to help them.
– Catch him, please catch him! – cried the policeman rushing on Hilbert Road towards the corner of Péter Street. He was almost out of breath.
Leo immediately followed him, running as fast as he could, barely touching the ground with his feet.
– Who are we supposed to catch? – said Mathina to a second police officer slowing down right beside her to catch a breath. The cop carried a remarkable assortment of old logical tools, among others a disguise, a tube voltmeter, and a level. These were more of a burden since they kept slipping off the man’s shoulders and of course they were heavy as well.
– There he is, that thief stole all the even numbers of this town! – he said gasping heavily.
Leo had already reached the corner, and quickly disappeared onto Péter Street. Mathina took a couple of bags off the policeman’s shoulder to speed up the chase. When they reached the Hilbert-Péter corner, they saw that Leo was only a few meters away from the thief.
The boy stepped forward, reached out for the man, and almost grabbed the stalk of the digit 4, which was sticking out of the bag – when the thief suddenly turned left onto Gödel Street. Leo could not slow down anymore, and found himself crashing into the policeman running from the other end of Péter Avenue.
– Let me help you – Mathina offered her hand to the policeman lying on the ground. As the logical equipment was collected, Mathina saw Leo already setting off on Gödel Road to catch the thief.
– Come back, you have no chance! – cried the policeman pulling himself together.
– I’ll help other cops, we’ll hunt him down together – Leo shouted, turning back.
– There are no more policemen! It’s only the two of us, one at both ends of the town! – the policemen looked at each other in despair. Leo stopped immediately. He was devastated. – What? Only the two of you? – The policemen nodded so emphatically that even Leo could see it from the distance.
They caught up with Leo, then at the meeting of five corners, turned left onto Hemachandra Street, and finally sat down in a cafe on Russell Square, the city’s main square.
– Why are there only the two of you? – Mathina was puzzled.
– Logi-city’s Chief of Police says two policemen are more than enough. Everyone else has been relocated to the cyber-security unit. Of course I understand that most crimes are committed in cyberspace, but we can’t bear this anymore here. I tried to convince him but he replied we are not doing our job well enough.
– I don’t think two people are able to patrol this city, – added Leo and bit into his Barcan-Marcus cake.
– Yes, this was our argument, too, – replied the other cop – but he doesn’t believe us.
– Show us the map of the city, please. We are in Logi-city, aren’t we? Maybe the chief can be persuaded by the power of logic – said Mathina.
One of the policemen pulled out a yellowish paper from his pocket and unfolded it.
Mathina and Leo were perplexed to see the city on such an outdated medium. The policemen explained that this illustrated how far the police had withdrawn from physical reality and that they barely receive any support. The outdated logical equipment was still available, but it was entirely useless.
Mathina took a long look at the map.
– So, the question actually is the following… Is it possible to place two cops in the city in such a way, that they are able to survey all of the streets, – she thought to herself.
Help Mathina and Leo find a good plan by moving 2 policemen on the map.
App: you can experiment by moving 2 policemen on the map: http://www.rimosco.it/ew/LC_S3_CatsDogs/index.htm?v=1
– Of course, two cops are not enough! – Mathina exclaimed.
– Of course, we know that that’s not enough. We experience it. But could you please prove this to the Chief, too, Miss? – One of the policemen asked. Mathina gave him a murderous look.
– If you never call me Miss again, I will. – Leo laughed and patted Mathina’s shoulder, while one of the cops nodded almost in fright.
– So, look, here are these three horizontal streets [Euler, Hemachandra, Péter]. They are parallel and have no intersection. Now, if you control the first horizontal road and your colleague the second, then neither of you will be able to survey the third, for sure! Of course, because you can’t find any spot in the city from where you can see two parallel streets!
The cops were fascinated by Mathina’s reasoning. From experience they knew that two people were not enough, yet they couldn’t explain it logically.
– I found another explanation! – Leo boasted and pushed the last bite of the cake into his mouth.
– Now tell me, you really smart guy! – said Mathina expecting that Leo would say something stupid.
– Look, gentlemen. How many streets does Logi-city consist of? You were standing on Hilbert Road, while your colleague on Péter Street. The thief turned into Gödel Street, we turned into here, Hemachandra Street. Barcan-Marcus Street is perpendicular to this, while Constance-Jones Street crosses both. Ladd-Franklin is perpendicular to Constance-Jones and then there is still the Euler Street on the northern border of the city. This is, no matter what, eight roads or streets. Now, let’s check the intersections. Whichever we take, there are no more than three roads crossing. If we could find two intersections with three different roads crossing each, even then you could only keep watch over a maximum of six of them! But there are eight roads in town!
Mathina nodded. She knew that Leo wasn’t as stupid as she thought. The fact that in the vast majority of cases he acts before thinking, doesn’t mean that he is not able to think.
– You are both fantastic! – the policemen were amazed. – Would you please tell us how many policemen we actually need as a minimum? It would be good to approach the Chief of Police with a concrete proposal.
– If not two, then three! – announced Leo.
App: Help Mathina and Leo find a good plan by moving 3 policemen on the map (the app confirms the right solution). http://www.rimosco.it/ew/LC_S3_CatsDogs/index.htm?v=2
– We can’t tell this yet, – Mathina said. – At least three. – If we find a good plan, in which three cops patrol all the streets, then three really will be enough.
Mathina and Leo snucked the sugar cubes from the cops’ coffee plates and launched the experiment.
Leo simply pulled the cubes from one place to the other. – With brains, please! – Mathina ordered. – Do you remember? We should start out from the three parallel roads. – Mathina put a cube on each of the three horizontal lines and they started pushing and pulling them along the lines.
– I got it! – cried Leo. And he showed the map to the policemen:
They looked at the arrangement for a second or two and nodded with appreciation.
– That’s really convincing, thank you, – they swiped off the sugar cubes from the paper, then noticed that they actually couldn’t see anymore where they were. Mathina, seeing how clueless they were, pulled out a pen and marked the locations.
– Thank you, thank you! – they were so grateful.
Then suddenly, the thief who stole all even numbers of the city, appeared in the cafe window, noticed them, and fled again.
– Oh, we really shouldn’t be sitting here, when thieves are running across Logi-city! – the other policeman claimed.
– Sometimes it’s better to sit down and think, believe me – Mathina winked at them.
***
[You can make further experiments with randomly generated maps: http://www.rimosco.it/ew/LC_S3_CatsDogs/index.htm?v=3]