What 12 numbers in a C(12,2) and C(12,3) combination produce non-repeating sums?

Question

this is my problem:

I need 12 distinct numbers, could be something like (3,9,15,17,27,etc.) that when added up either in combinations of 2 or 3, the sums do not repeat.

For example, if the numbers are 1-12, this is an issue because 2+3 and 1+4 both = 5

What 12 numbers would guarantee non-repeating sums?

Thanks!

There is no unique set of such #s, e.g. $3^0, 3^1, 3^2$ .... e.g. would be another such set. $2^0, 2^1, 2^2,$.... is the simplest. — true blue anil, Jul 24 '15 at 10:47
@trueblueanil I believe the powers of two do not work. Because adding three numbers is also allowed. Powers of three would work. — wythagoras, Jul 24 '15 at 14:42
@wythagoras: I don't see why they shouldn't. Think of the numbers in binary form, whatever # of combinations you take, the # will be unique. With powers of 3, you can add and subtract, and still get unique positive integers, e.g, 1, 3-1, 3, 3+1, 9-3-1.... — true blue anil, Jul 24 '15 at 15:29
By the way, I asked this question as a guest before I verified and created my account...i'm unsure how to choose the best answer...? — Tbarca, Jul 24 '15 at 17:26
You can upvote using up arrows both answers and comments. And you can tick an answer to accept (i.e. it works for you). — true blue anil, Jul 25 '15 at 05:08
@true, I don't think a 1-rep user can vote. Tick, yes; vote, no. — Gerry Myerson, Jul 25 '15 at 06:28

true blue anil · Accepted Answer · 2015-07-25T05:10:49.717

Ok, just to consolidate.

1, 2, 4, 8 ,16,..... i.e. $2^0, 2^1, 2^2, 2^3, ... 2^11$ is the simplest sequence that will serve your purpose.

You can see that easily by writing all possible combos in binary form, viz. 1, 10, 11, 100, 101, ...

and observe that each is unique.

You can have other sequences. 3^0, 3^1, 3^2, 3^3, .... will obviously do, but interestingly, besides adding combos, you can also subtract appropriately to get 1, 2, 3,.....$3^{n-1}$ , e.g.

1, 3-1, 3 , 3+1, 9-3-1, 9-3, 9-3+1, 9-1, 9, ......

This fact is used in the puzzle of a traveller who has a 40-link gold chain, and must pay the inn-keeper 1 link a day for 40 days.

What 12 numbers in a C(12,2) and C(12,3) combination produce non-repeating sums?

1 Answers1