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.
Thus, the number in the cell (1000,100) is:
999 is equal to 1111100111 in binary notation.
99 is equal to 1100011 in binary notation.
Summing up (without carrying), we achieve:
1111100111
1100011
----------
1110000100
which is 900 in decimal notation.
Puzzle Thoughts
