Strange Patience Game

Solution 1:

Getting an average score of the number of suits doesn't surprise me. Intuitively, it takes $13$ cards to get the first ace, which uses up one suit full. Then $13$ more cards should get you a $2$, etc. Clearly this is a very rough argument. But when the number of suites gets very high you can only get a score of 13.

You can exactly calculate the chance of any given score, but it gets messier as you go up. To get $1,$ you have to have all the $2$'s before the first $1$. You can ignore the rest of the cards. If there are $n$ suits, the probability of this is $\frac {n!^2}{(2n)!}$. To get $2$, you need all the $3$'s before the first $2$ that is after the first $1$. You can enumerate the allowable orders of $1$'s, $2$'s, and $3$'s and calculate the probability.

Solution 2:

For small number of suits simulation confirms the hypothesis. Here is result for $n=4$ suits and $N=10^6$ trials:

But for greater number of suits it seems to become a local maximum, the largest value being 13. For $n=8$ and $n=10$ suits, $N=10^6$ it is

Solution 3:

Here is a remark which is too long to fit in the comments section.

Let us fix the number of suits $n$. Let $k$ be the number of cards in one suit. For any integer $m$, the probability that the final score is $m$ does not depend on $k$ for $k > m$. This is because if one gets a score of $m < k$, this means that the chain of cards stops at some point, and that depends only on the relative distribution of the cards corresponding to the $m+1$ first values. Higher values do not influence this event.

Going to the limit $k \to + \infty$, one gets a limit distribution for the final score (finite number of suits, infinite number of values) which depends on $n$. Then, one can wonder :

• what are the properties of this distribution? Is it finite with full probability? If so, how does it decay at infinity? What is its mean (if defined)? Its moments of higher order?

• How does it scales when one increases the number of suits $n$?

I think that these kind of question are related to what you ask, and that they may be more "natural" (or, at least, more likely to be tackled by usual methods).

PS : the limit $n \to + \infty$ at fixed $k$ produces a trivial limit, hence I took this order for the limits.