# Problem about a process with bins of balls [duplicate]

A friend of mine give me this problem for fun:

Given $\frac {n(n+1)}{2}$ balls, first we divide arbitrarily these balls in baskets, after that we make another basket with one ball of each basket e do this procedure infinitely.

I want to prove that one time this stabilizes with 1 ball in one basket, 2 balls in another basket, ..., n balls in another basket.

It seems easy to solve, he says we can use some concept of energy (???), I'm trying with some concepts of combinatorics without any success.

## marked as duplicate by joriki, user91500, Joel Reyes Noche, C. Falcon, ShaileshJul 7 '16 at 0:14

• Please give your question a meaningful title. – Chris Eagle Jul 8 '13 at 16:32
• @RossMillikan Then, the problem is, if, in the initial configuration, there were the entire $n(n+1)/2$ balls in the first basket of the first set, and none in the others, we do not run out of all the balls in the first basket of the first set of baskets after only "one pass." In fact, in this case the system will oscillate. – Lord Soth Jul 8 '13 at 16:52
• user85493, you're really going to have to explain this better. – dfeuer Jul 8 '13 at 16:53
• @LordSoth the number of basket is arbitrary. – user85493 Jul 8 '13 at 17:01
• @RossMillikan the initial condition is with an arbitrary number of baskets – user85493 Jul 8 '13 at 17:02

I believe what you are discussing is known as Bulgarian solitaire. This is a theorem of Jorgen Brandt, i.e., that the game ends as you have described when the number of balls (or cards) is a triangular number, i.e., of the form $n(n+1)/2$.

Here is a nice source to read over: (paywall)

Solution of the Bulgarian Solitaire Conjecture

Kiyoshi Igusa

Mathematics Magazine

Vol. 58, No. 5 (Nov., 1985), pp. 259-271