11 Feb 2013

Math-e-Monday: A Mouse Trap Strategy

Many of the problems found in mathematics competitions require some playing around before they can be confidently solved. Many such questions do not obey the school test paradigm of question-method-answer; many of them demand that you do a bit of thinking!

This week we’re going to look at some strategic games. Why, for example, is noughts-and-crosses (tic-tac-toe) a boring game? It is dull because once you’ve understood the strategy you know that you will only win against someone who is clueless! Two knowledgeable players will always draw... until they get bored and go do something else.

The games I like best are those that are open – in that you can see all the information – yet still challenging to solve. In terms of solitaire games, the classic Free Cell is a good example of an open game; there is no luck involved, if you fail to win, it’s your fault. (There is only one known insoluble deck in the Windows version.)

Writing a program to solve or win certain games can involve many lines of code. However, before the formal encoding, there will be some basic rules to follow; some dos and don’ts. Such informal methods are known as heuristics. These are useful when tackling seemingly complicated maths problems. It is very common to find a problem that appears to have a huge number of options, yet when you go through the rules and ‘play the game’ you find there are certain limitations that cut down the possible solutions to a manageable number that can then be enumerated.

But today is Monday and I’ve already gone through too many long words! So, here is a very simple game: Mouse Trap. It is an infuriating game and, when I’ve used it in schools, many students find it difficult to formulate a winning strategy. Your mission is to write out a heuristic method to win as many games as possible. This will largely be in English but you can use some mathematics if you find it easier. Such a syncopated style is often easier for a human to understand than lines of code.

There are four versions of Mouse Trap; the fewer the starting pillars, the harder the game. I would start with the original Mouse Trap and then move up or down depending on how difficult you find the game. Also, don’t worry so much about getting a High Score as that depends largely on the luck of getting an easy starting layout. The aim is to win, whatever the score.

Mouse Trap Senior – just 5 pillars (Hard!)

Mouse Trap – 9 pillars (Original)

Mouse Trap Junior – 10 pillars (Easier)

Mouse Trap Junior II – over 20 pillars (Way Too Easy!)

The Question

Write out a winning strategy for one of the Mouse Trap games.

Also, try playing this as a 2-player game with a friend. You’ll have to print out a hexagonal grid and use one of the random initial layouts from the online game. If you go to Incompetech, I set the grid to A4 paper, line weight 0.6 points and hexagon size 0.4 inches. This gives the required 11 hexagons across but too many going down so just black out the extra rows. The game board online is 11 by 11 hexagons.

Feel free to write your results below in the comments box. Is there a ‘fair’ game where two human players have an even chance of winning?

Have Fun!