# zbMATH — the first resource for mathematics

The computer solves the three tower problem. (English) Zbl 0778.60050
Consider the following probability problem: We have $$n$$ piles (towers) of $$x_ 1,\ldots,x_ n$$ chips, respectively. Each second a pile $$X$$ is selected at random, then another pile $$Y$$ is chosen at random and a chip is moved from $$X$$ to $$Y$$. Stop, when a) one pile becomes empty, b) $$n-1$$ piles become empty. Find the expected times $$f(x_ 1,\ldots,x_ n)$$ and $$g(x_ 1,\ldots,x_ n)$$ until stop, and the probability $$p_ i$$ that pile $$x_ i$$ wins.
The case b) was solved for any $$n\in\mathbb{N}$$. The result is $f(x_ 1,\ldots,x_ n)=\sum^ n_{x_ i<x_ k}x_ ix_ k,\quad p_ i=x_ i/(x_ 1+\cdots+x_ n).$ The case a) was solved for $$n=3$$ with the result $f(a,b,c)=3abc/(a+b+c).$ Exact computations for small $$a,b,c,d$$ indicate that the case $$n=4$$ is hopeless.
The paper touches a related problem: Players $$1,\ldots,n$$ start with $$n$$ chips. In one round each player stakes a chip. Then an $$n$$-sided symmetric die labeled $$1,\ldots,n$$ is rolled and the winner gets all the chips staked. Stop, when a) any player is broke, b) all but one player are broke. Find the times $$f$$ and $$g$$ until stop.
For $$n=3$$ the case a) was found 1966. It is $f(a,b,c)=abc/(a+b+c-2).$ The case b) was found to be $g(a,b,c)=ab+bc+ca-2abc/(a+b+c-2).$ For $$n>3$$ there is no solution up till now. Exact computations for small pile sizes indicate that the results do not grow as fast as in the case of the tower problems.

##### MSC:
 60G50 Sums of independent random variables; random walks
games of chance
Full Text: