From New Scientist #2404, 19th July 2003
First, draw a chessboard. Now number the horizontal rows 1, 2, …, 8, from top to bottom and number the vertical columns 1, 2, …, 8, from left to right. You have to put a whole number in each of the sixty-four squares, subject to two conditions:
1. Rows 1, 2, 3, 4, 5, 6, 7, 8 are equal to columns 3, 6, 4, 4, 1, 6, 8, 6, respectively;
2. If N is the largest number you write on the chessboard then you must also write 1, 2, …, N-1 on the chessboard.
The sum of the sixty-four numbers you write on the chessboard is your total.
1. What is the largest total you can obtain?
If you look at your chessboard with the numbers on it you will find that every column is equal to a row. Now imagine we are considering chessboards of all sizes.
2. Is it possible to find an n×n chessboard, with a number in every square, so that every row equals a column, but there is at least one column which does not equal a row? If so, what is the smallest n for which it is possible?