Y

8	37	78	29	70	21	62	13	54	5
7	6	38	79	30	71	22	63	14	46
6	47	7	39	80	31	72	23	55	15
5	16	48	8	40	81	32	64	24	56
4	57	17	49	9	41	73	33	65	25
3	26	58	18	50	1	42	74	34	66
2	67	27	59	10	51	2	43	75	35
1	36	68	19	60	11	52	3	44	76
0	77	28	69	20	61	12	53	4	45
	0	1	2	3	4	5	6	7	8

       Magic Squares and Labyrinth seed Forms                                                                                                                                                                                                                                  

In the Bridges 2025 workshop, we were given a method of     constructing an odd order magic square and shown its relationship to a labyrinth seed form.  We used the example of a 9 x 9 square.  We were left with the question of why the magic square and labyrinth were so related.  I will not address that question here.                                  

 What I will do is give a formula for the entries in the magic square. We number the rows from bottom to top, 0 to 8, and the columns, left to right, 0 to 8, and regard the rows and columns and diagonals as wrapping around (so as to make a torus). One successively enters into the cells all the numbers from I to 81, beginning at the cell directly below the centre cell and proceeding along the diagonal that                                                                                                             X

slopes down to the right, until it fills in all 9 cells of the diagonal,  and then one moves down two cells, and continues along that diagonal, and so on, yielding the magic square at the right.                                      

Let f (x , y) be the entry in the cell, (x, y).

Let MOD (A, B) be the remainder when A is divided by B.

f (x. x + 1) = MOD (36 – 9 x, 81)          (with some abuse of notation, cell (4, 5) is taken to be 81 instead of 0)

Beginning at cell (x, x + 1), and moving up column x, two cells at a time reduces the entry each time by 10 Mod 81. Thus, 

f (x, x + 1 + 2k) = MOD (36 – 9x – 10 k, 81)

Setting y = MOD (x + 1 + 2k, 9) and solving for k, yields:

 f (x, y) = MOD (36 - 9x – 10 * MOD (5(y – x – 1), 9), 81).

One can easily generalize this magic square construction and this formula to any odd order square.  There are many interesting patterns in this array of numbers.

Martin Levin

mdlevin_public@msn.com

 

                                                                                                                                               

 

 

 

 

                                                                      

 

    

 

 

       

 

 

 

 

 

 

 

 

 

 

 

 

 

The Labyrinth Society website

 Here's the link to the very helpful and interesting website of the international Labyrinth Society. They
hold an annual labyrinth gathering that I'd like to attend one of these years. 

There are lots of excellent links to films, papers, maps and discussions here.

Interesting films about magics squares and labyrinths from our presentation and beyond

 The Mathologer on the Korean king's method of constructing odd-order magic
squares: https://www.youtube.com/watch?v=FANbncTMCGc

Gardner's Double Appleton labyrinth dance:https://www.youtube.com/watch?v=nXS-U0HZn1M

Susan & company's film about community labyrinthn workshops on Hornby Island (Bridges 2022 Short Film Festival link): https://gallery.bridgesmathart.org/exhibitions/2022-bridges-conference-short-film-festival/susan-gerofsky

A video of a talk by John Conway on magic squares from the 2014 Gathering for Gardner (related to his 2015 Bridges talk, which was unfortunately not recorded): https://www.youtube.com/watch?v=ZK-RXLJT7eM

Slides from our 2025 Bridges Eindhoven workshop

 Here's a link to a PDF of the slides!

Examples of magic squares -- planetary and non-planetary

 Here's a link to the 33 pages of (random) magic squares that we used in our colouring activity in the
workshop.

It's interesting to note that there doesn't yet exist a general magic square generator online. You can find some generators, but they keep reproducing the same squares according to some particular algorithm. Considering that there are so many possible magic squares of different dimensions, it seems as though there ought to be a generating program available.  We think that there is also a great need for  comprehensive ways of categorizing all the magic squares to help us understand how they work and what patterns they embody!

Some very good resources for further exploration of labyrinths and magic squares

 

Some recommended books on labyrinths. 

• The Jeff Saward book is an excellent overview, and is the source of the world map of labyrinths and mazes used in our slides. 
• The Hermann Kern book is a huge compendium of information about labyrinths -- and also contains Robert Ferré's claim about magic squares and labyrinth seeds (as does the Ferré book!)

Here is the reference list from our Bridges 2025 workshop paper. Sorry that the type came out so dark here, but you can check these links on the Bridges 2025 papers archive link to our workshop paper at https://archive.bridgesmathart.org/2025/bridges2025-581.html#gsc.tab=0