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.
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.
Rectangles of type magic, associative, pandiagonal, and concentric can be made with program MagicRectangles.
Bordered magic rectangles can be made with program BorderedRectangles.
Note: To reduce picture width, .5 has been removed from all these bones numbers.
Note: To reduce picture width, .5 has been removed from all these bones numbers.
Note: To reduce picture width, .5 has been removed from all these bones numbers.
Note: To reduce picture width, .5 has been removed from all these bones numbers.
Note: To reduce picture width, .5 has been removed from all these bones numbers.
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