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Cutting 4 by $n$ grids into two congruent pieces

Robert Dougherty-Bliss, Natalya Ter-Saakov, Doron Zeilberger

Abstract

In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be solved by the transfer matrix method, and demonstrate this by computing the generating function for the number of ways to divide a 4 by n rectangle into two connected, congruent parts.

Cutting 4 by $n$ grids into two congruent pieces

Abstract

In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be solved by the transfer matrix method, and demonstrate this by computing the generating function for the number of ways to divide a 4 by n rectangle into two connected, congruent parts.
Paper Structure (5 sections, 1 theorem, 15 equations, 1 figure)

This paper contains 5 sections, 1 theorem, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The grammar
  3. Maple packages and asymptotics
  4. Future work

Key Result

Theorem 1

Let $c_n$ be the number of ways of cutting a $4 \times n$ grid into two (connected) congruent pieces. The (ordinary) generating function of $c_n$ is The coefficients have the (approximate) asymptotic expansion

Figures (1)

  • Figure 1: Finite state machine constructed to recognize $4 \times n$ Graham matrices. Rectangular states are start states, purple states accept if the string has an even number of symbols, and green states accept no matter what.

Theorems & Definitions (2)

  • Theorem 1
  • Definition 1