Forty-two pieces, numbered from 1 to 42, are disposed in a 6x7 grid, so that each pair of consecutive numbers are in cells that touch each other horizontally or vertically, not diagonally. Below, there are two partial drawings of the grid, where only 3 numbers appear, but only one of the two is correct.

+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
|    | 11 | 20 |    |    |    |    |    |    | 11 | 20 |    |    |    |    |
+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |
+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |
+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    |    | 31 |    |    |    |    |    |
+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
|    | 31 |    |    |    |    |    |    |    |    |    |    |    |    |    |
+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
|    |    |    |    |    |    |    |    |    |    |    |    |    |    |    |
+----+----+----+----+----+----+----+    +----+----+----+----+----+----+----+
          (first drawing)                         (second drawing)

Justify what is the correct drawing and complete its filling.

Well, If I understand correctly, the numbers from 1 to 42  form a long chain, snaking around the board, where the start and end of the chain do not need to join.  Necessarily, even-numbers are surrounded horizontally and vertically by odd numbers, and odd numbers by even numbers.  So there must be an odd number of cells between 11 and 31.  So the first drawing is correct.

Posted by Steve Herman on 2005-11-30 08:37:00