4
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Put the numbers $1, 2, 3, ..., 16$ in circles so that the sum of the four numbers on each side of the triangle should be equal.

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    $\begingroup$ Did you create this puzzle or did you find it somewhere? If you did not create it, please add a citation to where you found it $\endgroup$ – lioness99a Jan 17 at 9:39
  • $\begingroup$ @lioness99a, i found a paper with the sketch only, I tried the find the original. $\endgroup$ – Nick Jan 17 at 10:15
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A solution (probably many more of them exist):

(Sorry for the image quality)
Here, all 6 sides of 2 large triangles give the sum of 34.

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  • $\begingroup$ Definitely many more exist - e.g. from your solution there's no restriction on swapping 8 and 9, 12 and 16, 1 and 14. In addition 6 can be swapped with 11 and 2 with 15 if the corresponding pairs of circles above them are also swapped left-to-right. Those swaps could be done on ANY valid solution, so any others will spawn at least 2^5 = 32 "equivalent" solutions from swapping. $\endgroup$ – Steve Jan 17 at 10:46
  • $\begingroup$ How you find the 34? I found the sum of given numbers, it is 136, then 136 : 4 = 34. What is 4 here? $\endgroup$ – Nick Jan 17 at 11:39
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    $\begingroup$ Not only "many" more solutions, but many thousands of them... ignoring the "no-computers" tag (so won't submit an answer), I found a total of 27086 unique solutions, each of which corresponds to one of a set of 32 solutions that can be generated by swapping some of the circles. Totals along sides of the triangles can be anything from 30 to 38. $\endgroup$ – Steve Jan 17 at 12:01
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    $\begingroup$ @Nick 4 of the 16 numbers are totalled for each of the sides. To explain how totals other than 34 are possible - note that 8 of the circles are included in 2 totals, and 8 are included only in 1 total. $\endgroup$ – Steve Jan 17 at 12:11
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    $\begingroup$ Also note - a solution with total 30 can be converted to one with total 38 and vice-versa etc. by swapping each number (N) with the number (17-N). @trolley813 : 30 and 38 are reachable (I get 426 * 32 solutions for each), 29 and 39 are not. Nick: I have a total of 27086 * 32 solutions available. I don't think it's fair on those honouring the no-computers tag to share them. $\endgroup$ – Steve Jan 17 at 13:13

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