If it helps, the first problem (pairs that add up to 81) looks like Gauss's add up the first n integers problem looked at from a different angle (
http://mathcentral.uregina.ca/QQ/dat...02.06/jo1.html ). When asked to add up the numbers, he recognized that he could add up 1+80, 2+79, etc. In your case, you see that you are trying to find the combinations that add up to n+1, which will be the same combinations. So 40=80/2 should work (at least for the case of n is even like 80).
The 4 combination case might be similar. I found it interesting that average for pairwise was 81, where the average for groups of 4 is 162=81*2=81*4/2. Does that contribute anything meaningful to developing the algorithm? The average for groups of 3 is 121.5=81*3/2. Does that suggest anything.
How much help do you need with this? Developing an algorithm and proving that it works (without input from someone who has already fully explored this problem) could be fairly involved. I have looked at it a little bit, and it is an intriguing problem, but I am not sure I am prepared to commit the time needed to develop and prove a complete and rigorous algorithm for this. If I were to take on the problem, I would probably shrink it down (maybe work on the numbers 1 to 10 instead of 1 to 80). Then I would look for patterns in the "subset sums" to see if that suggested something to me.
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