Friday, October 22, 2010

A Brief Introduction to Mathematics Related to Games

Our presentation was basically a brief introduction to mathematics related to games. It did not involve the actual theoretical explanations of games using mathematics but provided an insight as to how we may approach to explain games and look for gaming strategies using mathematics.

Throughout the presentation it was emphasized that we only need elementary mathematics to explain many gaming strategies, yet, obviously problem solving skills and mathematical thinking should be there to come up with insights so as to model the game. Concepts like abstraction, iteration and recursion used in mathematical modelling was introduced. Within the limited time frame problems related to knight tours, domination and independence in chessboards, tiling, spatial arrangements and etc. were discussed.

Abstraction is simply the process of discarding the unnecessary details of a physical problem and re-writing the physical problem in mathematical terms (i.e. equations, symbols, diagrams and etc.) This concept was introduced using the Guarini's Problem which is the oldest known chess board problem dating back to 1507 AD. It was observed that by abstraction we can transform the original problem in to a diagram from which the result to be proved was trivial. (You can find some interesting chessboard problems here.)

What is the minimum moves  required to switch
the position of the white and black knights ?  

Concept of abstraction was further applied to solve famous chessboard tiling problems making use of polyminoes. (i.e. Gomory's theorem and more) 3-D Constructions of cuboids were also a part of the exploration. Soma cubes was also briefly discussed.

7 Soma cubes and constructing a cube using them.

The concept of recursion was introduced using the towers of Hanoi problem. The solution provided at the presentation was a recursive one and it was emphasized that when dealing with games, recursive strategies are also important. 

Those who are interested can refer to these books where mathematics related to games are extensively discussed.

(This is the summary of what we discussed at our third session)

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