Three mathematical puzzles

 This is another selection from my book Playwiths, available from Amazon or through Polymoth Books. The apparent supplier is really just me, trading as Polymoth Books, but I set the firm up so I can supply booksellers and libraries more cheaply (note that some conditions apply).

This bit is free.

The crossed house puzzle

Your task looks simple: draw the diagram below by putting your pencil down on the paper, and drawing a single continuous line. You are not allowed to draw over any of the lines.

There is a solution, and in time, you will see a pattern!

You can solve this with a lot of difficult trial and error, or you can be mathematically clever, and work out a basic principle that applies to problems like this one and the next two as well. That’s a hint!

The question you have to ask yourself is this: “how many times do I enter or leave from one of the key points?” There is something very special about the points with odd numbers of starting and finishing points. The rest is up to you, but the problem does have a solution.

The prisoner and the cells

A prisoner in a rather strange prison (with even stranger guards!) was told that if he could find a way to walk through all of the doors of all of the cells, once and once only, he would be allowed to go free. The diagram below shows how the cell doors were arranged.

The prisoner’s puzzle.

* Image by en:User:Booyabazooka - http://en.wikipedia.org/wiki/Image:15-puzzle.svg, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1059593

Analyse the problem and see whether it is possible, and if it is, work out the solution. If it is not possible, prove it. Well, if you had any sense, you would have done the crossed house problem first. And that’s a hint. If you draw this figure on a torus in such a way that the hole of the torus is inside the middle cell at the bottom, it might be a bit easier—and that’s another hint.

The Königsberg bridge problem

In the city that was once called Königsberg, there were two islands in the river, linked to each other and to the shore by bridges as you can see in the diagram. The river is blue and the bridges are white. The problem for the citizens of Königsberg was this: was there any way of walking around the city and crossing each of the bridges once and once only?

 

A map of ancient Königsberg, with two islands in the river, and seven bridges.

Well, if you had any sense, you would have done the prisoner and the cells problem first. And that’s the last hint, for now. Now a research question: were/are there islands and bridges like this in Königsberg?

Notes

The three problems all have a common theme: entries and re-entries to certain points. If a cell in the second problem has an odd number of doors, you must either start inside it, or you must end in it, but not both. In the Königsberg bridges problem, each island has an odd number of entry and exit points, as does each bank. There is no solution to the second and third problems.

To find out about Königsberg reality, look for maps of Königsberg online.