magicrect

Construct magic rectangles and nearly magic rectangles of any order p × q for which one exists.

🔗 Live calculator: https://arijitray2.github.io/magic-rectangles/ — the same algorithms, running in your browser.

A magic rectangle of order p × q arranges the integers 1, …, pq so that every row sums to q(pq+1)/2 and every column sums to p(pq+1)/2. It exists precisely when p and q have the same parity — excluding the impossible 2 × 2 and single-row/column orders (Hagedorn 1999).

When p and q have opposite parity the magic constants are not integers, so a magic rectangle cannot exist; a nearly magic rectangle (Chai, Singh & Stufken 2019) exists instead: it uses 1, …, pq once each, has constant sums along one direction, and sums along the other direction differing by at most 1.

To our knowledge this is the first R package that constructs magic rectangles and nearly magic rectangles (the CRAN package magic covers magic squares and hypercubes only).

Installation

# install.packages("remotes")
remotes::install_github("Arijitray2/magic-rectangles")

Usage

library(magicrect)

magic_rectangle(3, 5)     # odd x odd magic rectangle
magic_rectangle(8, 10)    # even x even magic rectangle
magic_rectangle(6, 7)     # nearly magic rectangle (6 even, 7 odd)
magic_rectangle(7, 11)    # reproduces Chai-Das-Midha (2013), Section 3

rectangle_type(4, 7)      # "nearly magic"
rectangle_type(2, 2)      # "none" - the classical exception

r <- magic_rectangle(13, 19)
r$matrix                  # the 13 x 19 integer matrix
r$row_sums                # all equal q(pq+1)/2
verify_rectangle(r)       # "magic" - checked from first principles

Example output:

> magic_rectangle(3, 5)
3 x 5 magic rectangle

     [,1] [,2] [,3] [,4] [,5]
[1,]    1    9    7   10   13
[2,]    8   11   14    2    5
[3,]   15    4    3   12    6

Row sums:     40 40 40
Column sums:  24 24 24 24 24

What is implemented

Order p × q	Object	Construction
both even	magic rectangle	De Los Reyes, Das, Midha & Vellaisamy (2009)
both odd	magic rectangle	Chai, Das & Midha (2013): pattern matrix G_p, within-column interchanges, and the Theorem 2.1 composition for q ≡ 0 (mod 3)
even × odd	nearly magic rectangle	Chai, Singh & Stufken (2019): Theorems 2.1, 2.3, 2.4
2 × 2	—	provably impossible
1 × n, n > 2	—	provably impossible

Correctness

tests/exhaustive.R (run automatically by R CMD check) constructs every order 1 ≤ p, q ≤ 40 and verifies each result from first principles: the entries are exactly {1, …, pq}, and the row/column sums satisfy the relevant definition. The construction also reproduces the worked 7 × 11 and 13 × 19 examples printed in Chai, Das & Midha (2013) digit for digit. The JavaScript port used by the website is additionally cross-checked to be byte-identical to the R output on all 843 constructible orders with p, q ≤ 30, and self-verified up to 100 × 100.

References

1. T. R. Hagedorn (1999). Magic rectangles revisited. Discrete Mathematics 207, 65–72.
2. J. P. De Los Reyes, A. Das, C. K. Midha, P. Vellaisamy (2009). On a method to construct magic rectangles of even order. Utilitas Mathematica 80, 277–284.
3. F.-S. Chai, A. Das, C. Midha (2013). Construction of magic rectangles of odd order. Australasian Journal of Combinatorics 55(1), 131–144.
4. F.-S. Chai, R. Singh, J. Stufken (2019). Nearly magic rectangles. Journal of Combinatorial Designs 27(6), 368–376.