There is at least one riddle (or meaning) in each corner, a total of eight puzzles to solve, or readings to unveil. The 4x4 queens problem is not considered as such (it's solved anyway).

On the right, white plays, what would you do for both situations? There are 128 grains of wheat under the h1 knight.

For the domino, we ask the question: "Suppose a domino covers 2 squares of the chessboard, is it possible to cover the chessboard with dominoes such that only 2 diagonally opposite corners are not covered?"

For the other shape, the question is: "Can we cover the chessboard with such shapes (using 3 squares) in a way that exactly one corner (e.g. a1) is still free?"

If you can do it mentally, the question is raised to "Is the same possible (dominoes or "L" shapes) on a 16x16 board, or a 32x32?"

The final (9th) riddle is simple: how many squares are in a 8x8 standard chessboard?

If you read this box, it will spoil all riddles. But some answers are encrypted to avoid reading the solution too easily. There may be some hints in between.

______________ 1 ______________

Here is a hint for the checkers riddle: thisisthereversedhinteotcatcitisthereversedhint. It is pretty obvious with the big "X" actually...

Answer: Thesolutionisreversedkoonihcisthereversedsolution. No unbeatable player has been designed for chess. However last year only, the question has been addressed and solved for checkers. A breakthrough in AI which now has a deterministic algorithm that never loses. You can have a draw, but you can't win. The Way to Mastership perfect checkers playing is achieved by avoiding remembering all possible configurations. Only very simple games can be ideally played unless one is smart enough not to CHECK everything.

______________ 2 ______________

The back-right white position can win over black even though the situation seems symmetrical. The slight advantage for the white is playing first and being closer to the promotion. By sacrificing two pawns , the white player is able to promote before black survivors even though there are three of them. White can win for sure by pushing first the central pawn. Whatever pawn is played by the black (while grabbing the white pawn or moving forward), white only has to push the opposite one. In all resulting situations, black has the choice of grabbing two white pawns for a total of three (all of them), however, black can't grab both in one move. Therefore, the last white survivor is guaranteed to promote.

______________ 3 ______________

For the front-right white position, the hint is reversed:enonietam.

The solution: wasalreadyinthecorneraswhitecanmateinonebypromotingintoaknight.

______________ 4 ______________

The front-left 3x3 puzzle is well... puzzling. Of course, it refers to the 4x4 board problem where no queen can reach another. The hint was once again in the corner, as it is obvious that the queens are all at a "knight distance" from each other. There are only two but symmetrical solutions for this 4x4 subproblem. But this is not the actual reading I wanted to propose. The actual one is rather related to the 5x5 grid from the 3x3 (a1 to c3), not from the 4x4 board. The knight is the solution because if one puts one knight and one queen together on the same square, right in the center of a 5x5 board, they will complement each other and cover all the 25 squares. The center of this 5x5 board is the meanest square the title was referring to.

______________ 5-6 ______________

Both knights are also used for another dual problem. If the knight goes from a1 to h1 directly square by square, doubling the number of grains at each step, it would yield 128 grains. This leads to the well-known wheat riddle coupled with the knight-tour problem: what could be the path of the knight such that all 64 squares are visited exactly once. The amount of wheat would be gigantic of course although it may look like a very small reward at first. Here is one variation of the formulation.

* The ruler, who was not strong in math, quickly accepted the inventor�s offer, even getting offended by his perceived notion that the inventor was asking for such a low price, and ordered the treasurer to count and hand over the wheat to the inventor. However, when the treasurer took more than a week to calculate the amount of wheat, the ruler asked him for a reason for his tardiness.

* The treasurer then gave him the result of the calculation, and explained that it would be impossible to give the inventor the reward. The ruler then, to get back at the inventor who tried to outsmart him, told the inventor that in order for him to receive his reward, he was to count every single grain that was given to him, in order to make sure that the ruler was not stealing from him.

The Way to Mastership again.

______________ 7 ______________

The 6-4 domino solution may look really easy, but the original problem states a 8x8 GRID, not a chessboard. When one doesn't think of coloring the grid, one just can't answer the question for sure. Otherwise, it is easy to conclude that N dominoes covers exactly N white squares and N black squares, therefore there is no way to deal with two opposites corners as they must share the same color.

______________ 8 ______________

The "L" shape problem has a quick solution for the 8x8 board using two "L" covering 3x2 squares. The remaining 2x2 element needs a final "L". But when the board has no quick (and maybe lucky) solution, one must have a proof for any board size provided that the number of squares is 2^n X 2^n, for n > 0.

Suppose that we can cover the a1 to d4 4x4 board with "L" shapes such that a1 is still free. Then we can choose any rotation of that solution for the three remaining 4x4 boards. Therefore the 8x8 board can be covered by filling the three 4x4 boards such that e4, e5, and d5 are still free. We simply need a last "L" shape to complete these 3 free squares. Voil�.

Since solving the problem on a 8x8 board was a success, then we can instantly declare that covering a 16x16 board is also possible. We have a solution for the 8x8 board? Then we only have to cover the entire 16x16 with 4 times this solution, but by rotating it each time such that squares outside the board namely "i8", "i9" and "h9" are still free. We put the final "L" there, and the a1-h8 8x8 board has the original orientation, so a1 is still free.

______________ 9 ______________

How many squares are in a chessboard? There are 64 1x1 squares, 49 2x2 squares, 36 3x3 squares, etc... and 1 8x8 square, yielding 204 squares. There are also 1296 rectangles.