Dulan Edirisinghe’s Chess Corner: Logic and imagination, mathematics and chess – nation.lk – The Nation Newspaper

It is not a surprise that chess players tend to develop a natural affinity for mathematics, purely because imagination and logic play a major role in both disciplines. But unlike some other universally popular games such as poker or scrabble where the mathematics of probability plays a huge role, chess does not deal with chance or incomplete information. Hence in Game Theory, chess is called a Perfect information game because each player, when making a decision, has access to all the information.

The problem with such perfect information games is they nosedive into irrelevance if the game is solved. From Tic-tac-toe, which a schoolboy can solve, to Checkers, which took eighteen years of hard work to solve, so many strategy games have suffered this fate. The bottom line is, in this age of supercomputers with their astonishing processing power, the clock is ticking for most perfect information strategy games. What about chess then?

Why has chess not been solved yet

Chess playing computer engines are way stronger than any human. But still, we are a long way away from knowing the absolute truth in every position. Why? Because of the sheer complexity of the game.

Back in the 19th century it was found that there are more than 300 billion ways that the first four moves of a game could be played. Later, Claude Shannon, the father of information theory, argued that while it is theoretically possible to solve chess, it wont be feasible because it would take 1090 years to evaluate all the relevant positions.

The magic of geometric progression

Wheat placed on the first five squares (photo Wikipedia)

Such gigantic numbers and chess go back a long way if you go by the ancient story about the maharaja and his courtier who supposedly invented the Chaturanga, the earliest form of chess. Legend has it that the maharaja was so pleased with this new game that he offered any reward to his courtier.

The wise man asked for a single grain of wheat for the first square of the chessboard. Then two grains for the second, four for the third, and so on. Doubling each time, until the 64th and final square is reached.

The ruler has apparently laughed it off as a meager prize for such a brilliant invention. Soon to his shock and dismay, he was informed by his treasury that all the wheat grown in the world would not be enough to fulfill the courtiers request. The unbelievable final number, they correctly calculated, is 18,446,744,073,709,551,615 (264-1).

Versions differ as to whether the smart courtier was beheaded or promoted. The ruler, by all accounts, was either very offended or very impressed.

Some Mathematical tasks

As we mentioned earlier, logic and imagination goes hand in hand when it comes to chess and mathematics. Here are a couple of interesting problems to put your skills to the test.

After 16.Nxb1 Find a proof game from the starting position (Philip Lehpamer)

Logic tells us that White must make 15 captures. We also know that white cant capture anything with the first move. Therefore from moves 2-16, white must make fifteen consecutive captures.

Somehow this task reminds one of the game Peg Solitaire (more commonly known in Sri Lanka as Chuck-a-blass thanks to a local tele drama which made the game popular in the 1990s).

The second problem involves symmetry which is an important concept in many branches of mathematics, especially when it comes to geometry. One of Sam the Puzzle King Lloyds best known problems features a unique copycat problem.

Black copies each of whites first three moves. Then white delivers a checkmate with his fourth move. Find the moves.

Loyds solution was 1.d4 d5 2.Qd3 Qd6 3.Qf5 Qf4 4.Qxc8#. A similar solution can be arrived at by playing 3.Qh3 Qh6 as the third move.

Lloyds first solution

Can you find an alternative solution?

Solutions

1.Chuck-a-blass problem

1.Nc3 b5 2.Nxb5 c6 3.Nxa7 Nf6 4.Nxc6 Ne4 5.Nxb8 Nc3 6.Nxd7 Ba6 7.Nxf8 e5 8.Nxh7 Rf8 9.Nxf8 g6 10.Nxg6 Bb5 11.Nxe5 Ra3 12.Nxf7 Qd6 13.Nxd6+ Ke7 14.Nxb5 Nb1 15.Nxa3 Ke8 16.Nxb1.

2.Checkmate the copycat problem

1.c4 c5 2.Qa4 Qa5 3.Qc6 Qc3 4.Qxc8#

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Dulan Edirisinghe's Chess Corner: Logic and imagination, mathematics and chess - nation.lk - The Nation Newspaper