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On Card guessing after a single shelf shuffle

Markus Kuba

TL;DR

The paper analyzes the number of correctly guessed cards after a single shelf shuffle under full feedback by deriving a distributional recurrence for $X_n$ and solving it with generating functions, yielding an explicit $S(z,v)$ and moments. It proves a central limit theorem with $\mathbb{E}(X_n)=\frac{3}{4}n$ and $\mathbb{V}(X_n)=\frac{n}{16}$, and extends the framework to an asymmetric shelf shuffle with parameter $p$ and to a refined split into pure luck $L_n$ and certified correct $C_n$, including their joint moments and a multivariate CLT. The results provide a complete probabilistic picture of the shelf-shuffle guessing game, along with robust methods applicable to biased shuffles and refined performance metrics. These contributions deepen the understanding of shelf-shuffle dynamics and offer a foundation for related sampling, randomization, and statistical testing scenarios.

Abstract

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is shelf-shuffled exactly one time. One after the other a single card is drawn from the shuffled deck. The guesser makes has guess and the card is shown until no cards remain. We provide a distributional analysis of the number of correct guesses under the optimal strategy. We re-obtain the previously derived expectation and add a complete description of the distribution. We also obtain a central limit theorem for the number $n$ of cards tending to infinity. Furthermore, we discuss an unbalanced, biased shelf shuffle and show how to derive the extend our analysis, also adding the complete position matrix. Finally, a refined analysis of the number of correct guesses is carried out, distinguishing between pure luck guesses and certified correct guesses.

On Card guessing after a single shelf shuffle

TL;DR

The paper analyzes the number of correctly guessed cards after a single shelf shuffle under full feedback by deriving a distributional recurrence for and solving it with generating functions, yielding an explicit and moments. It proves a central limit theorem with and , and extends the framework to an asymmetric shelf shuffle with parameter and to a refined split into pure luck and certified correct , including their joint moments and a multivariate CLT. The results provide a complete probabilistic picture of the shelf-shuffle guessing game, along with robust methods applicable to biased shuffles and refined performance metrics. These contributions deepen the understanding of shelf-shuffle dynamics and offer a foundation for related sampling, randomization, and statistical testing scenarios.

Abstract

We consider a card guessing game with complete feedback. An ordered deck of cards labeled up to is shelf-shuffled exactly one time. One after the other a single card is drawn from the shuffled deck. The guesser makes has guess and the card is shown until no cards remain. We provide a distributional analysis of the number of correct guesses under the optimal strategy. We re-obtain the previously derived expectation and add a complete description of the distribution. We also obtain a central limit theorem for the number of cards tending to infinity. Furthermore, we discuss an unbalanced, biased shelf shuffle and show how to derive the extend our analysis, also adding the complete position matrix. Finally, a refined analysis of the number of correct guesses is carried out, distinguishing between pure luck guesses and certified correct guesses.
Paper Structure (12 sections, 13 theorems, 53 equations)

This paper contains 12 sections, 13 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Shuffling techniques and card guessing games
  3. Notation
  4. Shelf Shuffle
  5. Distributional analysis
  6. Recursive description
  7. Distributional analysis
  8. Expectation, variance and limit law
  9. Extensions
  10. Asymmetric shelf shuffle
  11. Refined analysis of the ordinary shelf shuffle
  12. Outlook and summary

Key Result

Lemma 1

Suppose a deck of $n$ cards is shuffled once in a shelf-shuffler with one shelf. Let $m_{i,j}$ be the probability that card $i$ lands in position $j$. Then, we have with the convention that $\binom{0}{0} = 1$ and $\binom{0}{k} = 0$ for all positive integers $k$. In particular, $m_{i,j} = 0$ if and only if $1 \le j \le n-i$, and $m_{i,n-j+1} = m_{i,j}$ for all $i,j$.

Theorems & Definitions (24)

  • Definition 1: Single-shelf shuffle
  • Lemma 1: Position matrix of a single shelf shuffle
  • proof
  • Proposition 1: Optimal strategy for full feedback
  • Example 1
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • ...and 14 more