The "quadrata subtractionis" were published by Adam Kochański (Poland) in 1686.
The old paper "Considerationes quaedam circa Quadrata et Cubos Magicos"
was rediscovered by Henryk Fukś (Canada) and translated from Latin to English.
In his paper "Magic squares of subtraction of Adam Adamandy Kochański"
Henryk Fukś explains these special magic squares and some theory.
The following graphic is shown in this paper.
How to calculate the subtractions?
Choose any row, column, or diagonal of a square.
For example the first row of the first square: 1, 6, 13, 2
Sort the numbers in decreasing order: 13, 6, 2, 1
Add the arithmetic symbols "-" and "+" alternating: 13 - 6 + 2 - 1
Calculate the arithmetic term: 8
Kochański called the result residuum.
In all above shown squares the residuum is 8 for each row, column and diagonal.
Question: Is it possible to avoid the sorting of the numbers?
Yes, for certain squares this can be done by coloring the entries and postulate:
Take the positive difference of numbers with the same color and add all differences.
Here is an example of a magic 4x4-square of subtraction:
Build the positive differences of the red numbers and of the blue numbers in a certain row, column, or diagonal.
Then add these differences und you will obtain the residuum 8. This also is true for the semi-diagonals.
Moreover all rows and 4 broken diagonals are magic for addition.
(All odd numbers are in column 1 and 2, all even numbers are in column 3 and 4.)
Question: Are squares possible that are magic for subtraction and also for addition?
No such squares can be found for the orders 4 and 5. But several such squares exist for order 6.
Magic 6x6-square of subtraction (residuum = 15) and addition (sum = 111)
Say pd(red), pd(green), and pd(blue) are the positive differences of two entries in red, green, blue cells respectively. For each row, column, and diagonal: pd(red) + pd(green) + pd(blue) = 15
Question: How could magic squares like the above 6x6-square be found?
We considered all 32134 magic series of order 6 and chose only those which also are magic for subtraction with residuum 15.
From these we took only those where the two highest numbers are greater than 24, the two lowest are smaller than 13
and the remaining two numbers are greater than 12 and smaller than 25. There are 2025 suitable series.
Then we used a bit-vector program to find the 6x6-squares. The best program for order 6 was created by Hidetoshi Mino (Japan),
see magicsquare6.net (external link).
We obtained 1933 essentially different 6x6-squares that have the wanted properties.
In May 2024 Jos Luyendijk searched in the list of all bimagic order-8 squares
and found more than 600 magic squares of subtraction with residuum 32.
Here is a sample that additionally is associated (centrally symmetric).
Associated bimagic 8x8-square of addition that is a magic square of subtraction with residuum 32.
