[Paper Review] The Mondrian Puzzle: A Connection to Number Theory

Cooper O'Kuhn|arXiv (Cornell University)|Oct 10, 2018

Art, Technology, and Culture1 references4 citations

TL;DR

This paper establishes a nontrivial lower bound on the number of integers $ n \leq x $ for which the Mondrian Puzzle's minimal area difference $ M(n) \neq 0 $, using number-theoretic techniques despite the problem's geometric origins. It proves that the density of such $ n $ is at least $ \frac{C_\epsilon \log \log x}{\log x} $, with $ C_\epsilon = \frac{1}{e^\gamma (2\log 2 + \epsilon)} $, providing strong evidence that $ M(n) = 0 $ likely never occurs.

ABSTRACT

We obtain partial progress towards answering the question of whether the quantity defined in the Mondrian Puzzle can ever equal 0. More specifically, we obtain a nontrivial lower bound for the cardinality of the set $\{n\leq x: M(n) eq0 \}$ where $M(n)$ is the quantity appearing in the Mondrian Puzzle and $x$ is the usual quantity that one thinks of as tending to infinity. More surprisingly, we do so by use of number theoretic techniques in juxtaposition to the innately geometric nature of the problem.

Motivation & Objective

• To investigate the density of integers $ n $ for which $ M(n) \neq 0 $, where $ M(n) $ is the minimal area difference in the Mondrian Puzzle.
• To determine whether $ M(n) = 0 $ can ever occur, a long-standing open problem in combinatorial geometry.
• To establish a quantitative lower bound on the size of the set $ \{n \leq x : M(n) \neq 0\} $ using number-theoretic methods.
• To bridge the gap between the geometric nature of the Mondrian Puzzle and deep results in multiplicative number theory.

Proposed method

• The proof begins by identifying a subset of $ n $ satisfying a number-theoretic condition on divisors of $ n^2 $, which guarantees $ M(n) \neq 0 $.
• It uses the indicator function $ T_j(n^2)P_j(n) $ to encode the condition that $ \tau_2(n^2) = j $ and all divisors $ d \mid n $ satisfy $ d > j $.
• The sum is restricted to square-free $ n $, leveraging the asymptotic density $ \frac{6}{\pi^2} $ of square-free integers to preserve the lower bound up to a constant factor.
• The problem is reduced to counting $ r $-tuples of distinct primes $ p_1 < \cdots < p_r $, all greater than $ 3^r $, with product $ \leq x $, for $ r \leq \log_3(g_\epsilon(x^2)) $.
• The Prime Number Theorem and Mertens’ theorem $ \sum_{p \leq x} \frac{1}{p} = \log \log x + O(1) $ are used to estimate the number of such prime tuples.
• The final bound is derived by summing over $ r $, leading to a lower bound of order $ \frac{x \log \log x}{\log x} $, with explicit constant $ C_\epsilon $.

Experimental results

Research questions

• RQ1 Can the set $ \{n \leq x : M(n) \neq 0\} $ be shown to have positive density as $ x \to \infty $?
• RQ2 Is it possible to prove that $ M(n) = 0 $ never occurs for any $ n $, using analytic number theory?
• RQ3 What is the best possible lower bound on the size of $ \{n \leq x : M(n) \neq 0\} $, and how does it relate to the distribution of divisors of $ n^2 $?
• RQ4 Can the structure of $ \tau_2(n^2) $ be used to rule out the existence of a perfect Mondrian tiling for any $ n $?
• RQ5 What role do square-free integers and prime factorizations play in bounding the number of potential solutions to the Mondrian Puzzle?

Key findings

• The paper establishes a nontrivial lower bound: $ \left|\{n \leq x : M(n) \neq 0\}\right| \geq \frac{C_\epsilon x \log \log x}{\log x} \left(1 + O_\epsilon\left( \frac{\log \log x}{\log x} \right) \right) $, where $ C_\epsilon = \frac{1}{e^\gamma (2\log 2 + \epsilon)} $.
• The bound implies that the natural density of $ n $ with $ M(n) \neq 0 $ is at least $ \frac{C_\epsilon \log \log x}{\log x} $, which tends to zero but slowly.
• The proof relies on a number-theoretic reduction: $ \{n \leq x : \forall d < n^2, d|n^2 \Rightarrow d \tau_2^*(d) < n^2 \} \subseteq \{n \leq x : M(n) \neq 0\} $, which is derived via contradiction from area constraints.
• By restricting to square-free $ n $, the analysis simplifies significantly, and the sum is reduced to counting $ r $-tuples of distinct primes $ > 3^r $ with product $ \leq x $, for $ r \leq \log_3(g_\epsilon(x^2)) $.
• The method uses the Prime Number Theorem and Mertens’ theorem to estimate the number of such prime tuples, yielding the stated lower bound.
• The authors conclude that their method cannot achieve a bound of the form $ \epsilon x $ for any $ \epsilon > 0 $, indicating the need for new ideas to resolve the conjecture that $ M(n) = 0 $ never occurs.