Start by writing all the numbers from 1 to 15 as a list.
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
November 2022 Clayton McDaniel
Can you rearrange the numbers 1 to 15 (1-n) such that the sum of any adjacent number is a square number?
Then I thought to find all the square numbers that were even possible with 15 numbers
14 + 15 = 29 so any square number that is below 29 is a possible choice.
4,9,16, or 25
After some thinking and fiddling around on paper, considering the connections between numbers, I noticed this...
I didn't notice the connection at first but my relatively wavey lines made me think of something flowing.
Do you see it? After noticing that only 2 numbers had single connections (8 and 9), I started tracing the lines and found the path!
The connections form the path that reveals the sequence. 9,7,2,14,11,5,4,12,13,3,6,10,15,1,8
I then extended this idea to the other known solutions 17,16,23,25
I struggled and failed to find a solution with 26 numbers even though there is one.
I did notice something else after I found the sequence.
There is an underlying pattern of squares.
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
9,7,2,14,11,5,4,12,13,3,6,10,15,1,8
(9 + 7 = 16) (7 + 2 = 9)...
16,9,16,25,16,9,16,25,16,9,16,25,16,9
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
16,9,7,2,14,11,5,4,12,13,3,6,10,15,1,8
(16 + 9 = 25) (9 + 7 = 16)...
25,16,9,16,25,16,9,16,25,16,9,16,25,16,9
1,2,3,4,5,6,7,8,9,10,11,12,13,14,1516,17
16,9,7,2,14,11,5,4,12,13,3,6,10,15,1,8,17
(16 + 9 = 25) (9 + 7 = 16)...
25,16,9,16,25,16,9,16,25,16,9,16,25,16,9
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23
18,7,9,16,20,5,11,14,22,3,1,8,17,19,6,10,15,21,4,12,13,23,2
(18 + 7 = 36) (7 + 9 = 16)...
36,9,25,36,25,16,25,36,25,4,9,25,36,25,16,25,36,25,16,25,36,25
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25
18,7,9,16,20,5,4,21,15,10,6,19,17,8,1,3,22,14,11,25,24,12,13,23,2
(18 + 7 = 36) (7 + 9 = 16)...
36,16,25,36,25,9,25,36,25,16,25,36,25,9,4,25,36,25,36,49,36,25,36,25
Notes
- A proof was found to the Square-Sum Puzzle by Robert Gerbicz detailed in this video. His proof was posted on this thread.
- Robert Gerbicz's proof shows that a pattern exists but is far more complex than what I'm seeing. My simple graphs are more likely the start of something that will be just as complex. Probably a deadend though...as I'm sure everyone found these connections also.
- Another pattern also emerged after I drew the lines. It seems that the most common square number possible, with each list, plays the biggest role. It connects most of the numbers right away. Then, if you start from the largest number, working your way backwards connecting any left-over numbers with 1 or 0 connections, you will eventually connect the paths. There seems to be a bit of fiddling around with the smaller digits so this is probably a false pattern. Plus, these few possible insights have not helped in finding a path beyond 25.