[GFunk911]

[ QUOTE ]
[ QUOTE ]

Any 12-sum will have a sum of 12C + xA, where C is the smallest gear, A is the above constant and x is some number. In addition, each gear has the weight C + xA.

[/ QUOTE ]

You're using the same x for both the sum and for an individual gear, as far as I can tell, which is problematic.

[/ QUOTE ]

Sorry, they are different x's. Probably should have used x and y to signify.

Addendum: I didn't prove that oyu can't do it with different weights. Here's more

Say you have 12 of the weights in groups of 6 so they equal. The 13th weight is A more than a weight in the 12. You swap them. You not cannot make the 2 6-groups equal, since this would require moving .5A from the swapped group to the unswapped group, and all your weights are multiples of A. The weight you swap in needs to be 2Ax more (or less) than the weight swapped out, meaning that your weights need to be in multiples of 2A.

Repeat problem with 2A in space of A, 4A in space of 2A, etc.

Better?