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Square pairs (Posted on 2005-01-09) Difficulty: 3 of 5
Back in An Arrangement of 15 you were asked to place the numbers 1 to 15 in a line so that any two adjacent numbers summed to a square number.

Now, try to arrange the numbers from 1 to 32 in a circle, so any two adjacent numbers again sum a square number.

See The Solution Submitted by e.g.    
Rating: 4.3333 (12 votes)

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beginning | Comment 1 of 12

The total of all 32 sums must be 31*32 = 992 since each number participates in exactly 2 sums.

The numbers 25 through 32 can each have only 2 possible neighbors since only 36 and 49 are permissible sums that are greater than 25.  Thus the following "mini sequences" must appear in the final answer, though perhaps in reverse order:
4 32 17
5 31 18
6 30 19
7 29 20
8 28 21
9 27 22
10 26 23
11 25 24
These account for 16 of the 32 sums.  The total of these 16 sums is 680.
The remaining 16 sums must add up to 312.
Finding the number of ways to add 16 numbers from the set {4, 9, 16, 25, 36} such that the sum is 312 may limit the number of possibilities of other "mini sequences"
Furthermore, there are limited numbers of these squares:
There is only 1 "4", 4 "9"s, 7 "16"s, 12 "25"s, and 6 "36"s remaining.
One way is:
1 "4", 4 "9"s, 4 "16"s, 4 "25"s, and 3 "36"s  which gives 16 numbers adding up to 312.  There may be others.

Edited on January 9, 2005, 5:05 pm
  Posted by Larry on 2005-01-09 16:23:48

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