BDi - Clueless receive accolade from Berkeley maths Professor

The following is the full text of a letter of support received from Professor Jorge Nuno Silva of Lisbon and Berkeley Universities, and well known author and games expert.....

The word “Sudoku” is a short version of a Japanese expression that means something like “the numbers should be single”.

Ïn Sudoku the fact that we use numbers is irrelevant. We could as well play with colors or letters for the same purpose. This is a purely logic puzzle. Because it is so purely logic and uses numbers, some take it for a mathematical exercise, which is not true.

This puzzle was created by Howard Garnes, a retired architect when, at 74, published in the American journal Dell Pencil Puzzles and Word Games, under the title “Number Place”. Later, in 1986, it was published with huge success in Japan. In 1997, a retired judge from New Zealand, Wayne Gould, noticed one of these puzzles. He spent the following years developing a software program able to generate new Sudoku puzzles at the level of the ones produced by the Japanese, which were based on human creativity. Gould’s puzzles made it into western newspapers, showing up in The Times in 2004. From then the diffusion was fast and today many newspapers carry daily puzzles.

The Japanese company Mikoli, responsible for the introduction of the game in Japan, holds that its puzzles are better made than the ones depending on the computer. The process of generating a good puzzle is more an art than a technique. Gould argues that, for each level of difficulty, the puzzles produced by his software are more elegant and carry less clues at the starting position. The readers of Times seem to find him right.

Even though it is not an arithmetic puzzle, Sudoku has roots in some objects that dear to the mathematicians.

In the 18th century, the Swiss mathematician Euler studied the so called latin squares, which are square arrays of numbers, nxn, in which each row and each column carries the numbers 1, 2, …, n. Each Sudoku puzzle successfully filled is, of course, a latin square of order 9x9. But, as the Sudoku has a special restraint relative to the nine boxes, there are more latin squares 9x9 than Sudoku puzzles. To be exact, there are

5 524 751 496 156 892 842 531 225 600 latin squares 9x9 and 6 670 903 752 021 072 936 960 Sudoku puzzles. Two numbers hard to apprehend…

Euler was led to his research on latin squares when he was studying magic squares, which are square arrays of numbers in which the sums of the rows, columns and main diagonals are the same. Here is an example of a 3x3 magic square:

8	1	6
3	5	7
4	9	2

As can be easily checked all rows columns and the two diagonals add up to 15.

Sudoku shows then a very rich pedigree. As usually happens with good puzzles, there are some theoretical aspects of Sudoku that are still far from being well understood. It is hard to characterize the different difficulty levels of the puzzles, beyond considering the amount of clues given at the outset. How can we guarantee keeps the difficulty level along its resolution, avoiding sudden trivialization upon filling of a certain cell? When making up a new puzzle, how can we be sure its solution is unique?

Some of these questions are addressed by brute force computer calculations, but not all are within reach of actual machine performance. The Japanese have a point when they prefer human conception over computerized one.

Sudoku is, to say it in few words, an excellent logic puzzle. As it uses numbers, it is not restricted, as cross words are, to any particular cultures, which helps to explain its immense universal success.

However, it is, as we usually say, a game for one player. The usual variants vary the dimensions and form of the array. But, the ones proposed by BM Clent and D Shenton go much further, giving a new flavor to this puzzle. Clicstrips is a good example: the obligation of treating the numbers in groups of three and the absence of initial clues changes the nature of the mental activity, by introducing a new complexity.

However, we still have puzzles…

But the same authors created games for two players from the the puzzles. Clent and Shenton transformed a good puzzle in an excellent game for two people. Actually, in several such games.

Their variants make it possible to associate Sudoku with all the drama of a good boardgame, always respecting Sudoku’s original essence. As a matter of fact, the way they build the games bring attention to the most interesting features of the original puzzle. The new approach avoids the solitary way it was usually practiced, offering us excellent social recreation. It is only fair to emphasize that this recreation has high intellectual level. Shendoku and Ninja Squares are great examples!

As any good game, chess for example, Clent and Shenton’s games work in several levels of sophistication, always with interest, being appropriate for people of the age range 7-77, just like Tim Tim.

Jorge Nuno Silva

University of Lisbon.