Draw only one continuous line into the 8x8 diagram below, which begins in the left cell above (with a “01”) and ends in the right bottom cell (with a “33”). The line must pass through all the numbers from 01 to 33 (through the center of each cell), once and only once through each one, and can run only horizontally or vertically (not diagonally). Besides, the line may not cross itself.

Far from being a trial and error problem, this can be solved logically.

     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 01  | 16  | 05  | 20  | 25  | 09  | 21  | 01  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 18  | 10  | 27  | 26  | 11  | 17  | 12  | 32  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 32  | 11  | 15  | 19  | 08  | 06  | 27  | 20  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 17  | 04  | 13  | 24  | 30  | 28  | 31  | 02  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 25  | 10  | 02  | 26  | 04  | 28  | 22  | 13  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 05  | 14  | 30  | 08  | 15  | 31  | 19  | 06  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 23  | 07  | 24  | 16  | 29  | 22  | 18  | 19  |
     +-----+-----+-----+-----+-----+-----+-----+-----+
     | 03  | 12  | 09  | 03  | 07  | 14  | 23  | 33  |
     +-----+-----+-----+-----+-----+-----+-----+-----+

(In reply to Solution by ed bottemiller)

Ed, there was some trial-and-error, but I, too, approached it logically by noticing that the 21 and 29 occurred only once, the 19 three times and knowing the 01 in the upper right would not be on the path. Working from both ends of the path, I eliminated the squares that became isolated. The path requires two sides of each square, thus any square on the edge of the grid already had one side that the path could not take and each corner, two. When a square became eliminated from consideration, this lessened the number of sides the path could travel through adjacent squares and were then sometimes eliminated from consideration. Eventually the path connected.

I suspected there may be alternate routes. In fact, when I first observed Leming's solution, I immediately saw the alternative route you presented.

Posted by Dej Mar on 2008-09-23 23:17:10