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 |
+-----+-----+-----+-----+-----+-----+-----+-----+
I noticed that 21 and 29 occur only once (so must be on the path) and that 19 occurs three times; the 01 in the upper right cannot be used. About 20 could be directly located from the immediate neighbors (except for 19, any placement or exclusion allowed corresponding exclusion or placement of the other occurrence of that nuimber), though not necessarily showing a unique path. Perhaps it is possible to derive the entire path "analytically" but at that stage I just used trial and error to find the rest of the selected cells and the entire path.
My solution path, starting from the upper left is:
01,18,32,11,10,27,05,20,25,09,21,12
17,06,08,19,15,13,02,26,24,30,04,28
31,22,29,16,03,07,14,23,33
An interesting task -- especially if there is a way to find the entire path by sequential assignment/exclusions with no trial and error (I'll watch to see if any such is presented.)
Solution
