# Magic Rectangles

## Introduction

A **normal** magic rectangle of order **r** x **c** contains the integers
**1, 2, ..., rc** arranged such that the sum of the numbers in each row
is the same, and the sum of the numbers in each column is the same:

row sum: **c(rc + 1)/2**
column sum: **r(rc+1)/2**

There are no odd by even **normal** magic rectangles, because for these
(**rc + 1**) is odd; then, if **r** is odd the column sum is fractional and if **c** is odd the
row sum is fractional.

All unqualified references to magic rectangles on this site mean
**normal** magic rectangles.

## Pandiagonal

Diagonals are generally not considered in non-square magic rectangles.

However, if diagonals are as shown in the example below, there are 'pandiagonal' magic
rectangles for some orders. The diagonal sums are the same as the sums for the
shorter of the rows or columns.

## Construction

Rectangles of type magic, associative, pandiagonal, and concentric can be made with program MagicRectangles.

Bordered magic rectangles can be made with program BorderedRectangles.

## Magic Examples: odd

### Orders 3x5, 3x7, 3x9

## Magic Examples: even

### Orders 2x4, 2x6, 2x8

## Associative Examples: odd

### Orders 3x5, 3x7, 3x9

## Associative Examples: even

### Orders 4x6, 4x8, 4x10

Note: To reduce picture width, .5 has been removed from all these bones numbers.

## Near-associative Examples: even

### Orders 6x10, 6x14

Note: To reduce picture width, .5 has been removed from all these bones numbers.

## Pandiagonal Examples: odd

### Orders 5x7, 5x9, 5x11

## Pandiagonal Examples: even

### Orders 4x6, 4x8, 4x10

Note: To reduce picture width, .5 has been removed from all these bones numbers.

## Concentric Examples: odd

### Orders 5x7, 7x9, 9x11

## Concentric Examples: even

### Orders 4x6, 4x8, 4x10

Note: To reduce picture width, .5 has been removed from all these bones numbers.

## Bordered Examples: odd

### Orders 5x7, 7x9, 9x11

## Bordered Examples: even

### Orders 4x6, 6x8, 8x10

Note: To reduce picture width, .5 has been removed from all these bones numbers.

##### REFERENCES

Bhatt, Saurabh and Prof. Ashish Das "Construction of Magic Rectangles"

http://www.math.iitb.ac.in/~ashish/Magic/

Chai, Feng Shun, Ashish Das and Chand Midha

"Construction of magic rectangles of odd order"

http://www.math.iitb.ac.in/~ashish/Magic/paper.pdf

Danielsson, Holger "Book: Magic Rectangles"

https://www.magic-squares.info/info/book-rec.html

De Los Reyes, J. P., Ashish Das, Chand K. Midha and P. Vellaisamy

"On a method to construct magic rectangles of even order"

http://www.math.iitb.ac.in/~ashish/Magic/paper2.pdf

Taneja, Inder J.
ijtaneja@gmail.com

"Even Order Magic Squares Using Bordered Magic Rectangles"

https://numbers-magic.com/?p=4292