A page of postage stamps in a booklet contained 6 stamps from which it was possible to obtain any total value from 1 to the total value of all the stamps on the page by the appropriate selection of one or more orthogonally connected stamps from those laid out in the following manner:
+-----+-----+-----+
| | | |
| 1 | 2 | x |
| | | |
+-----+-----+-----+
| | | |
| 4 | 6 | y |
| | | |
+-----+-----+-----+
With the numbers chosen, four of the values could be obtained in two different ways each, and the rest could be obtained in only one way each.
Your job is to figure out the numeric values of x and y.
Note that the square portion already given, containing 1, 2, 4 and 6, would, by itself satisfy the mentioned criterion: all the values from 1 to 13 can be achieved as the total of orthogonally connected stamps, such as 1, 2 and 4 to make 7 (the 6 and the 1 are not orthogonally connected, so that would not be allowed).
Didn't we have this, or an almost identical, puzzle not long ago? It is not hard to find a solution, using the minimum values, i.e. x=5 and y=3. As I read the puzzle we need only be sure that all stamps used are attached to one another; the only ones which are excluded are those which attach to one of the others ONLY orthogonally (as would 1 and 6 alone, to represent 7).
1 (1) 2 (2) 3(3) 4(4) 5(5) 6(6) 7(1,2) 8(35) 9(36) 10 (46) 11(146 ) 12(246) 13(1246) 14(1346) 15(2346) 16(2356) 17(12356) 18(3456) 19 (13456) 20 (23456) 21 (123456)
Four alternatives: 8(26) 9(126) 10(235) 13(256).
However, this does not satisfy the condition that ONLY four numbers could be met in more than one way. (e.g. 14(356) 15(12345).
Solution
