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A mathematical chess problem is a problem formulated using a chessboard, sometimes of size different from the normal size, normal chess pieces and rules, and of a mathematical nature. For an example consider the following problem:
Example 1 What is the maximum number of non-attacking rooks that can be placed on an 8×8 chessboard?
(A) 6 (B) 8 (C) 10 (D) 12 (E) 14
Solution Since only one rook can be placed in a column and there are only 8 columns in a chessboard the maximum number must be less than or equal to 8. The following placing of rooks shows that this maximum can be achieved:
Therefore the answer is (B).
Example 2 Let A be the set of points in the plane such that x and y are both positive integers less than or equal to 20 and B be a subset of Asuch that no two points in B are at a distance of √5. What is the maximum number of elements in B?
(A)50 (B) 150 (C) 200 (D) 250 (E) 300
This does not look like a mathematical chess problem. But this problem can be made to look like a mathematical chess problem. If we consider a 20×20 chessboard then two squares that are at a distance of √5 (distance between the centers of the squares) are squares that can be occupied by attacking knights. So the number of elements in B is the maximum number of non-attacking knights on the 20×20 chessboard. So what is the maximum number of non-attacking knights that can be placed on a 20×20 chessboard? Since a knight always move from a black square to a white square and vice versa it is clear that knights placed on all white color squares or all black color squares are non-attacking. So 200 non-attacking knights can be placed on the chessboard. Is this the maximum? We can pair all the squares in the chessboard as shown below for 4 squares:
In this pairing only one knight can be placed in each pair of squares. If two are placed then they attack each other. There are 200 such pairs. So the maximum number of non-attacking knights is less than or equal to 200. So the maximum number of non-attacking knights is 200. Therefore the maximum number of elements in B is 200. Therefore the answer is (C).
Problem of the week #1
Let A be the set of points in the plane such that x and y are both positive integers less than or equal to 12 and B be a subset of A such that no two points in B are at a distance of √10. Which of the following is/are true?
- Number of elements in B could be 72.
- Number of elements in B could be 73.
- Number of elements in B could be 143.
(A) I only (B) II only (C) III only (D) I and II only (E) All
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