Engrave numbers on 5 keys on a circular keyring so that the numbers on adjacent groups of keys sum to any value between 1 and 21 inclusively.

For example, 1,1,3,6,6 can sum up to any number between 1 and 17 (1=1, 1+1=2, 3=3, 3+1=4, 3+1+1=5, 6=6, 6+1=7, 6+1+1=8, 6+3=9, 6+3+1=10, etc).

(In reply to Most of solution by Tristan)

But if the 2 is next to the 1 we cannot complete the circle when we try to place the 4. (i.e. if we put the 4 next to the 2, then we need a 5 somewhere other than next to the 1 (that would leave 2 sums of 6) and we'd need a 9 in between leaving no sum of 8) If we put the 4 next to the 1, then we need a 6 and an 8 which also fail.  And if the 4 is not adjacent to the 1 or 2 then we need a 5 and a 9 which also fails.

So we need a 1 and a 2, but not adjacent, and we need a 3 next to the 1.   A 5 between the 1 and 2 gives sums up to 9 as well as 11.  Add a 10 and we are done.

Solution: 3,1,5,2,10

Posted by Eric on 2005-03-28 23:44:36