The squares of an infinite chessboard are numbered successively as follows: in the lower left corner (first row, first column) we put 0 (zero), and then in every other square we put the smallest nonnegative integer that does not appear to its left in the same row or below it in the same column. See it partially filled:

            |   |   |   |   |   |   |   |   |
            +---+---+---+---+---+---+---+---+--
            | 5 |   |   |   |   |   |   |   |
            +---+---+---+---+---+---+---+---+--
            | 4 | 5 |   |   |   |   |   |   |
            +---+---+---+---+---+---+---+---+--
            | 3 | 2 | 1 |   |   |   |   |   |
            +---+---+---+---+---+---+---+---+--
            | 2 | 3 | 0 | 1 |   |   |   |   | 
            +---+---+---+---+---+---+---+---+--
            | 1 | 0 | 3 | 2 | 5 |   |   |   |
            +---+---+---+---+---+---+---+---+--
            | 0 | 1 | 2 | 3 | 4 | 5 |   |   | 
            +---+---+---+---+---+---+---+---+--

Find the law that rules the numbers that fills the chessboard, so that in seconds, you can evaluate the number that is, for example, in the intersection of the 1000th row and the 100th column.

I agree that XOR fills the chessboard as shown, but has anybody proved that XOR satisfies "in every other square we put the smallest nonnegative integer that does not appear to its left in the same row or below it in the same column"?

Posted by Bractals on 2005-10-29 16:35:59