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On cases where Litt's game is fair

Anne-Laure Basdevant, Olivier Hénard, Edouard Maurel-Segala, Arvind Singh

TL;DR

The paper investigates a generalized Litt's game where two words $A$ and $B$ of equal length define scoring in a sequence of fair coin flips. It demonstrates that the game is fair whenever $A$ and $B$ share the same auto-correlation, i.e., $[A|A]=[B|B]$, regardless of their inter-correlation, by constructing an explicit bijection that exchanges Alice and Bob's scores. This bijection is built via an involution $\phi$ on overlaps that preserves total counts, leading to equality of $\mathbb{P}_n(\text{Bob wins})$ and $\mathbb{P}_n(\text{Alice wins})$ for all $n$ and establishing a formal symmetry between the two players. The work connects pattern matching with probabilistic arguments, providing a rigorous mechanism for when score differences are symmetric and highlighting potential avenues for fixed-length asymptotics and broader conjectures about tie and pattern interactions in such games.

Abstract

A fair coin is flipped $n$ times, and two finite sequences of heads and tails (words) $A$ and $B$ of the same length are given. Each time the word $A$ appears in the sequence of coin flips, Alice gets a point, and each time the word $B$ appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt's game that recently set Twitter abuzz. We show that Litt's game is fair for any value of $n$ and any two words that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores; the fact that the inter-correlation does not come into play in this case may come up as a surprise.

On cases where Litt's game is fair

TL;DR

The paper investigates a generalized Litt's game where two words and of equal length define scoring in a sequence of fair coin flips. It demonstrates that the game is fair whenever and share the same auto-correlation, i.e., , regardless of their inter-correlation, by constructing an explicit bijection that exchanges Alice and Bob's scores. This bijection is built via an involution on overlaps that preserves total counts, leading to equality of and for all and establishing a formal symmetry between the two players. The work connects pattern matching with probabilistic arguments, providing a rigorous mechanism for when score differences are symmetric and highlighting potential avenues for fixed-length asymptotics and broader conjectures about tie and pattern interactions in such games.

Abstract

A fair coin is flipped times, and two finite sequences of heads and tails (words) and of the same length are given. Each time the word appears in the sequence of coin flips, Alice gets a point, and each time the word appears, Bob gets a point. Who is more likely to win? This puzzle is a slight extension of Litt's game that recently set Twitter abuzz. We show that Litt's game is fair for any value of and any two words that have the same auto-correlation structure by building up a bijection that exchanges Bob and Alice scores; the fact that the inter-correlation does not come into play in this case may come up as a surprise.
Paper Structure (2 sections, 2 theorems, 29 equations)

This paper contains 2 sections, 2 theorems, 29 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['bijection']}

Key Result

Theorem 1

Let $A$, $B$ two words of length $\ell$ such that $[A | A]= [B | B]$. Assume that, under $\mathbb{P}_n$, the letters $(\varepsilon_i)_{1\leqslant i\leqslant n}$ of $X_n$ form an i.i.d. sequence with the uniform distribution on $\{\text{H},\text{T}\}$. Then for each $n\geqslant 1$, $(N_A(X_n),N_B(X_n

Theorems & Definitions (7)

  • Theorem 1
  • Definition 2
  • Definition 3
  • Definition 4
  • proof
  • lemma 1
  • proof