[Paper Review] Magic squares with empty cells

Abdollah Khodkar, David Leach|arXiv (Cornell University)|Apr 30, 2018

Graph Labeling and Dimension Problems4 references3 citations

TL;DR

This paper introduces k-diagonal magic squares—n×n arrays with entries confined to k consecutive diagonals, each row and column containing exactly k filled cells, and all rows and columns summing to the same magic sum. The authors prove a k-diagonal magic square exists if and only if n=k=1 or 3≤k≤n with n odd or k even, using direct constructions for k=3 to 6 and a recursive superposition method based on shifting and scaling to extend solutions to arbitrary k.

ABSTRACT

A k-magic square of order n is an arrangement of the numbers from 0 to kn-1 in an n by n matrix, such that each row and each column has exactly k filled cells, each number occurs exactly once, and the sum of the entries of any row or any column is the same. A magic square is called k-diagonal if its entries all belong to k consecutive diagonals. In this paper we prove that a k-diagonal magic square exists if and only if n = k = 1 or 3 <= k <=n and n is odd or k is even.

Motivation & Objective

• To define and investigate k-diagonal magic squares, a new class of partially filled magic squares where entries are restricted to k consecutive diagonals.
• To determine the necessary and sufficient conditions under which such k-diagonal magic squares exist for given n and k.
• To provide explicit direct constructions for k-diagonal magic squares when k=3,4,5,6 and extend them to all valid k via superposition.
• To resolve the structural constraints arising from diagonal confinement and parity conditions on n and k.

Proposed method

• Define a k-diagonal magic square as an n×n array with exactly k filled cells per row and column, entries from 0 to kn−1, and constant row and column sums.
• Construct explicit solutions for k=3,4,5,6 using modular arithmetic to assign entries to specific diagonals, ensuring all numbers are used exactly once.
• Use a shifting operation (mod n) to generate new k-diagonal magic squares from existing ones, preserving the magic sum and diagonal structure.
• Apply a superposition technique: combine an ℓ-diagonal and m-diagonal magic square (with ℓ+m ≤ n) by shifting the second by ℓn and placing entries in non-overlapping cells to form an (ℓ+m)-diagonal square.
• Leverage additive scaling: adding a constant m to all entries of a k-diagonal magic square preserves the magic sum property and extends the range of entries.
• Prove existence for all valid k by expressing k as a non-negative integer combination of 3,4,5 (for odd n) or 4,6 (for even n), and applying the superposition lemma recursively.

Experimental results

Research questions

• RQ1 For which values of n and k does a k-diagonal magic square of order n exist?
• RQ2 What structural and arithmetic constraints arise from restricting entries to k consecutive diagonals in a magic square?
• RQ3 Can direct constructions for small k be generalized to all valid k using algebraic and combinatorial superposition techniques?
• RQ4 Why do parity conditions (n odd or k even) emerge as necessary for existence when k≥3?
• RQ5 How do the magic sum and entry distribution behave under diagonal confinement and superposition operations?

Key findings

• A k-diagonal magic square exists if and only if n=k=1 or 3≤k≤n with n odd or k even, establishing a complete characterization.
• Direct constructions are provided for k=3 (odd n≥3), k=4 (n≥4), k=5 (n≥5 odd), and k=6 (n≥6 even), each using modular indexing across k consecutive diagonals.
• The magic sum for any k-diagonal magic square of order n is k(kn−1)/2, which is consistent across all rows and columns in the constructions.
• The superposition lemma allows combining valid ℓ- and m-diagonal magic squares (with ℓ+m≤n) into an (ℓ+m)-diagonal magic square by shifting and scaling the second square.
• For odd n and k≥3, every k can be expressed as 3a+4b+5c for non-negative integers a,b,c, enabling construction via repeated superposition of the base k=3,4,5 cases.
• For even n and k≥4, every k can be expressed as 4a+6b for non-negative integers a,b, enabling construction via superposition of k=4 and k=6 base cases.