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.
What a fascinating result, pc! There is nothing in the problem statement that even remotely suggests anything about binary numbers or bitwise operations. Yet, the result is exactly that...a table showing the results of the bitwise XOR operation on any two integers--an operation that can only be performed by considering the binary representation of each number.
I'd love to see a proof.
Amazing
