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Starting with a pile of 10⁴ tokens, Bob plays the following game: at each round, he picks one of the existing piles with at least 3 tokens, takes away one of the tokens in this pile, and separates the remaining ones into two non-empty piles of arbitrary size. Bob stops when all piles have identical size. What is this size and what is the maximal number of piles?

First, Bob can easily reach a decomposition that prevents all piles to be of the same size: for instance, he can start with a pile of 1 and another pile of 2. Looking at the general perspective, an odd number of tokens, n=2k+1, can be partitioned into (1,1,2k-1). Which means that the decomposition (1,1,…,1) involving k+1 ones can always be achieved. For an even number, n=2k, this is not feasible. If the number 2k can be partitioned into equal numbers u, this means that the sequence 2k-(u+1),2k-2(u+1),… ends up with u, hence that there exist m such that 2k-m(u+1)=u or that 2k+1 is a multiple of (u+1). Therefore, the smallest value is made of the smallest factor of 2k+1. Minus one. For 2k=10⁴, this value is equal to 72, while it is 7 for 10³. The decomposition is impossible for 2k=100, since 101 is prime. Here are the R functions used to check this analysis (with small integers, if not 10⁴):

solvant <- function(piles){
if ((length(piles)>1)&((max(piles)==2)||(min(piles)==max(piles)))){
return(piles)}else{
i=sample(rep(1:length(piles),2),1,prob=rep(piles-min(piles)+.1,2))
while (piles[i]<3)
i=sample(rep(1:length(piles),2),1,prob=rep(piles-min(piles)+.1,2))
split=sample(rep(2:(piles[i]-1),2),1,
prob=rep(abs(2:(piles[i]-1)-piles[i]/2)+.1,2))
piles=c(piles[-i],split-1,piles[i]-split)
solvant(piles)}}
disolvant <- function(piles){
sol=solvant(piles)
while (min(sol)maxle){
lasol=sol;maxle=length(sol)}}
return(lasol)}