Group by Value

A pack of cards contains $4n$ cards with four identical cards of each value. The pack is shuffled and cards are dealt one at a time and placed in piles of equal value. If the card has the same value as any pile it is placed in that pile. If there is no pile of that value then it begins a new pile. When a pile has four cards of the same value it is removed.

Throughout the process the maximum number of non empty piles is recorded. Let $E(n)$ be its expected value. You are given $E(2) = 1.97142857$ rounded to 8 decimal places.

Find $E(60)$. Give your answer rounded to 8 digits after the decimal point.

This problem requires an understanding and application of probability and expected value. It involves some complicated calculations and is not typical of standard mathematics problems. Computing expected values for large scenarios like the one in question usually depends on creating a recursive formula or generating function since calculating directly can be overly complex.

Here is a brief plan for how we could theoretically proceed:

1). We need to calculate probabilities for each possible configuration of piles. We can map the number of piles onto a state space, where each state represents a different configuration of piles.

2). For each state, we calculate the probability of drawing a card to either 1) create a new pile, or 2) complete an existing pile. The sum of these two actions would be the total next state probabilities.

3). We’d multiply these probabilities by the number of piles remaining in each case to get the expected pile count in the next state, and sum these up to get the expected value.

4). To calculate $E(60)$, we would iterate through each state for $n=60$, sum up the expected values, and divide by the total number of states to get the average expected value.

However, performing these steps manually for $E(60)$ is impractical because of the large number of possible states and the complexity of the calculations involved. Therefore, this is a problem that would typically be solved via a computer program, using dynamic programming, exact re-summation or Monte Carlo simulation for instance.

The detailed math behind this solution involves a mix of combinatorics, probability theory, and dynamic programming optimization, which is not feasible to explain in a simple text form.

So, while I can’t provide the exact value for $E(60)$ without writing and running this complex program, I hope this gives you some insight into how the problem could be approached and solved.

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