Patterns with Magic Squares, Numbered Polyominoes and Polyomino Tilings

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This page looks at a (tenuous?) relationship between polyominoes and magic squares based on numbered polyominoes and a method of constructing magic squares attributed to De la Hire in W.S. Andrews' book "Magic Squares and Cubes".

The Magic Square Construction Method

The magic square construction method can be used for order 5 magic squares and higher prime order squares. It uses two subsidiary squares that each have a different set of five numbers arranged so that every row, column and diagonal (including broken diagonals) has five different numbers. The geometrical arrangement of the set of numbers in one subsidiary square is the reverse of the arrangement of the set of numbers in the other square.
+ =

Adding the numbers in corresponding cells of the two squares gives a magic square in which all of the rows, columns and diagonals (including broken ones) add to the same total of 65. The blue coloured cells in the subsidiary squares show that the geometrical arrangement of numbers in the two squares is reversed. The red coloured cells in the final magic square show an example of a broken diagonal.

Each of the 25 cells of the 5 by 5 square formed by superimposing one subsidiary square on top of the other has one of the 25 unique ways to select a pair of numbers taking one number of the pair from the set 1,2,3,4,5 and the other number from the set 0,5,10,15,20.
5,153,54,202,01,10
2,201,05,103,154,5
3,104,152,51,205,0
1,55,203,04,102,15
4,02,101,155,53,20

If each subsidiary square is repeated horizontally and vertically to generate two 'infinite' subsidiary grids of numbers (here only small subsets of the infinite grids are shown) then any 5 by 5 squares taken from each grid will produce an order 5 magic square.

The picture below shows the grids with the numbers removed leaving just the patterns formed by the repeated occurrence of a particular number in the grids. The pattern could be described as a series of knight's move diagonals. The two grid subsets also highlight the reverse arrangement of numbers. Again only small subsets of the two infinite grids are shown.

The Polyomino Connection

Now for the Polyomino Connection: numbered polyominoes can be used to generate the subsidiary grids used in the magic square construction method described above. Here a numbered polyomino is simply a polyomino with a number in each component square.

The integers from 1 to 5 have been used to number the polyomino below and the numbers 0, 5, 10, 15, 20 used to number the polyomino's mirror image:

The picture below shows a portion of the two tilings and the number grids generated from the polyomino and its mirror-image shape. There is no special significance to the colours used - several different colours have been used as an easy way to distinguish the individual polyominoes in the tiling. The number grids generated by tiling with the polyomino and its mirror image are the same grids used in the magic square construction method described above.

Which Polyominoes Can Be Used?

The example above shows one of the twelve order 5 polyominoes being used to generate the subsidiary grids for the magic square construction method. However not all of the twelve polyominoes can be used in this way. The first requirement is that the polyomino can tile in such a way that there are no gaps and all repetitions of the polyomino are oriented in the same way.

There are twelve order-five polyominoes if you don't count rotations and reflexions.

Some of these tile the plane in a regular way - using just identical copies of the same polyomino all oriented the same way. Below are two examples. In these pictures there is no special significance to the colours used - several different colours have been used only as a way to distinguish the polyomino shapes in pictures of tilings.


The polyominoes below can't be used to produce this special kind of regular tiling.

The nine polyominoes that can be used to create such a regular tiling can be split into two groups depending on the regular pattern that a particular polyomino square makes in the tiling.

Group 1

Group 2

Group 1 Polyominoes

In the first group, the pattern formed by a particular polyomino square in the tiling is a set of simple diagonals. The example below illustrates this using one of the polyominoes from Group 1 with an asterisk in the same square of each polyomino the tiling.

Sample Group 1 Polyomino

Sample Group 1 Polyomino Tiling

Group 1 Polyomino Grid Pattern

Group 2 Polyominoes

In the second group, the pattern formed by a particular square in the polyomino is a set of knight's move diagonals. The pattern has 90 degree rotational symmetry but has no mirror image symmetry. Again, this is illustrated using one of the polyominoes from Group 2 with an asterisk in the same square of each polyomino in the tiling.

Sample Group 2 Polyomino

Sample Group 2 Polyomino Tiling

Group 2 Grid Pattern

Polyominoes in Both Groups

The polyomino below appears in both groups - it can tile the plane in both the simple and knight's move diagonal patterns.

The straight line polyomino also appears in both groups - it can create an infinite number of different tiling patterns.

Creating the Magic Square Grids

Tilings with numbered polyominoes from the second group (the knight's move tiling group) can generate the subsidiary grids used in the magic square construction method. If the integers from 1 to 5 are used to number a polyomino from the second group and the numbers 0, 5, 10, 15, 20 are used to number the polyomino's mirror image:

The picture below shows a portion of the two tilings and the corresponding number grids generated from the polyomino and its mirror-image shape.

This is the heart of the polyomino/magic square connection: tilings with certain numbered polyominoes can generate the subsidiary grids used in the order 5 magic square construction method.

Completing the Journey from Polyomino to Magic Square

Now take any 5 by 5 square of numbers from each of the above subsidiary grids and add the numbers in corresponding cells together to get a magic square of order 5. Every row, column and diagonal (including the broken diagonals) has the same sum.

+ =

The Magic Square Tiling Test

An order five polyomino can tile the plane if it fits onto either the 'knight's move' diagonal grid or the simple diagonal grid so that there are five different symbols within the boundary of the polyomino. These grids use the integers from 1 to 5 but any set of five different symbols could have been used. A particular polyomino might have to be rotated and/or reflected when testing on one of the two grids.

Simple Diagonal Grid

The order five polyominoes that fit onto the simple diagonal grid so that each of the five polyomino squares has a different symbol are shown in the picture below.

Knight's Move Diagonal Grid

The order five polyominoes that fit onto the knight's move diagonal grid so that each of the five polyomino squares has a different symbol are shown in the picture below.

Non-Tiling Order 5 Polyominoes

The following pictures show order 5 polyominoes that don't fit onto either the simple or knight's move diagonal grid. All of the rotations and reflections (if appropriate) for each polyomino are shown superimposed on a 5 by 5 simple grid and a 5 by 5 knight's move grid to illustrate that these polyominoes can't be flipped or rotated so that they cover 5 different symbols.




Last Updated on February 23rd 2007

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