Chessboard Problems ExoticArithmetic III
1 Introduction
We begin with some definitions. A set of chess pieces is called independent if none of them attack any of the others. A set of pieces covers a board if among them they occupy or threaten each square. Let us denote the squares of an 8 × 8 board using matrix notation.
a11 a12 a13 a14 a15 a16 a17 a18
a21 a22 a23 a24 a25 a26 a27 a28
a31 a32 a33 a34 a35 a36 a37 a38
a41 a42 a43 a44 a45 a46 a47 a48
a51 a52 a53 a54 a55 a56 a57 a58
a61 a62 a63 a64 a65 a66 a67 a68
a71 a72 a73 a74 a75 a76 a77 a78
a81 a82 a83 a84 a85 a86 a87 a88
2 Knight Problems
Knight problems come in many types. There are two types of domination problems and two types of independence problems and they occur on finite rectangular boards, often 4 × 4 or 8 × 8. These are counting problems and probability problems. We offer two examples. A set of chess knights is called independent if none of them attack any of the others.
1. How many ways can 8 independent knights be placed on an 4 × 4 chess board? To facilitate the solutions and to offer a modest hint, we num- ber the squares of the 4 × 4 board so the odd numbers squares are white and the even ones dark. The second figure shows the number of squares attacked by a knight on the square.
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1 2 3 4 2 3 3 2 8 7 6 5 3 4 4 3 9 10 11 12 3 4 4 3 16 15 14 13 2 3 3 2
2. How many ways can 7 independent knights be placed on an 4 × 4 chess board?
3. How many ways can 6 independent knights be placed on an 4 × 4 chess board?
4. Prove that nine knights cannot be arranged independently on a 4 × 4 chess board.
5. Three knights are randomly placed on a 4 × 4 chessboard. What is the probability that none attack the others?
6. Note that a knight move from a position (i, j) to a position (u, v) in the plane, can be characterized by the equation |i − u| · |j − v| = 2. Equivalently, we can say (u − i)2 + (v − j)2 = 5. We can use the notation (i, j)N(u, v) if and only if |i − u| · |j − v| = 2 used in discussion binary relations.
(a) Characterize bishop moves using matrix notation. (b) Characterize rook moves using matrix notation. (c) Characterize king moves using matrix notation.
7. (a) Find the maximum number of independent rooks, and the number of ways to arrange them on a board. (If no mention is made of the board’s size, assume it is 8 × 8.) (b) Answer the same two questions for bishops. (c) Answer the same two questions for knights. (d) Answer the same two questions for kings.
8. Another important type question regards the number of paths of a knight or a king from one square to another. Now suppose the king
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is unrestricted. Now our piece is a chess king in the plane. We ask questions like how many paths of length 6 are there that start and end at the origin? We give two alternate solutions. Here is one of my favorites. 9. How many locations can a knight occupy after 10 moves if it starts at the origin on the plane. 10. (a) Find the minimum number of rooks required to cover a board, and the number of ways to arrange them on a board. (b) Answer the same two questions for bishops. (c) Answer the same two questions for knights. (d) Answer the same two questions for kings. 11. There are some famous old questions related to knight’s tours, covering the board by queens, and maximal sets of independent queens that are not appropriate for this course, but are worth mentioning here: (a) Find all ways to put eight independent queens on a chess board. (b) Find all ways to put n independent queens on an n × n chess board. (c) Find a covering of a chess board by five queens, and count the number of ways to do this. (d) Find a complete recurrent knight’s tour.
3 Interchange Problems
Our problem here is to interchange the three white knights with the three black knights in as few moves as possible. We do not require the black and white pieces to move alternately.
♥ ♥ ♥
♣ ♣ ♣
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4 Covering
A perfect cover (also called a tiling) of an m by n chess board by tiles (usually dominoes) is an arrangement of so that (a) the entire board is covered, (b) each tile covers a whole number of squares and (c) there is no overlap except along the edges.
12. Count the number of perfect covers of the chess boards listed below:
(a) 2 × 1, 2 × 2, 2 × 3, 2 × 4, 2 × 5, and 2 × 6. More generally, let Gn denote the number of tilings of a 2 × n board. Find Gn for n = 1, 2, 3,..., 10. The five perfect tilings of a 2 × 4 are shown
below:
(b) 3 × 3, 3 × 4, 3 × 5, and 3 × 6. (c) The 62 square board obtained by removing the opposite ends of one of the diagonals.
13. Generalize the definition of tiling to include solid blocks. Find the number of tilings of a 2 × 2 × 2 block by blocks of size 2 × 1 × 1
14. Is it possible to tile a punctured 8×8 chess board (one with one square removed) using regular triominoes ( )? Does it matter which square is removed?
15. Consider a 2n × 2n punctured board. Show by induction that such a board has a multiple of 3 number of squares. Can such a board be tiled by L-shaped triominoes ?
16. A knight is placed on an infinite chess board at the origin (0, 0). Let K(n) denote the number of positions the knight can occupy after n moves. For example K(1) = 8. Find K(n) as a polynomial function on n.
17. How many ways can four independent chess kings be placed on a 4 × 4 board?
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18. How many king’s paths of length 5 are there from the square numbered 1 to the square numbered 16? A path on length 5 can visit square 16, leave and then return, for example, 1, 6, 11, 16, 12, 16. How many paths of length 3 are there? Length 4?
19. Probability problems. For each two chess pieces randomly placed on two squares of a chessboard, find the probability that one of the pieces attacks the other.
(a) knight and rook. (b) bishop and queen. (c) knight and queen. (d) bishop and rook.
20. From Alex Soifer. Forty-one rooks are placed on a 10 × 10 chessboard. Prove that there is among them an independent set of five.
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