# 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.

## Construction

Magic rectangles, associative magic rectangles and concentric magic rectangles 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.

## 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

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