Franklin Squares

Contents

Introduction

Benjamin Franklin (1706-1790), the early American scientist, statesman and author, is known as the creator of bent-diagonal magic squares. (Actually these squares are not magic in the accepted definition because the two main diagonals do not sum correctly.)

In his lifetime he published an order-8, an order-16 and a magic circle. The order-16 and the magic circle were first published in Ferguson’s Tables and Tracts Relative to Several Arts and Sciences (London,1767). In the next few years, all three were published in various works and personal letters.

These three have been published many times since and will be reproduced here.

The orders 4, 6 and 8 will be reproduced below. A variation of the order-16 (by moving four columns from the left to the right sides) will also be shown, as well as a feature comparison of Franklin’s two squares and my modified one.

The paper mentioned above is Paul C. Pasles, The Lost Squares of Dr. Franklin: Ben Franklin’s Missing Squares and the Secret of the Magic Circle, The American Mathematical Monthly, 108:6, June-July, 2001, pp 489-511. It includes 49 citations.

I extend my thanks to Paul C. Pasles, for bringing this exciting news to the attention of recreational math buffs. Much of the information on these pages was derived from his paper. See also, his Web page at http://www.pasles.org/Franklin.html .

Donald Morris has an attractive new site (February 2006) on Franklin Squares. He explains what he thinks is Franklins method of construction.
He also explains his own method and shows some excellent examples.
This excellent site is at http://www.bestfranklinsquares.com/

Franklin`s Order-8

square

Because of wrap-around, these patterns may be started on any of the squares 64 cells. These two patterns may so started on any odd numbered column so can appear in 32 positions. (The vertical version of these don't work.) The bent diagonals and half row and columns work only if started on columns (or rows for vertical) so appear 16 times in the square.

The well known 16 x 16

This square was published during Franklin’s lifetime (in 1767), and has been published many times since. 
It contains most of the same patterns that appear in his order-8 square, plus many more. It contains the bent diagonals which are a trademark of his squares. However, like his order-8, it is not considered a true magic square because the main diagonals do not sum correctly (one sums 128 too low, the other 128 too high).

This square will be compared later with the newly discovered square when we look at the many embedded patterns.

NOTE: This square can be made magic (i.e. correct main diagonals) simply by rotating the top right quadrant clockwise and the bottom right quadrant counterclockwise. However, this results in the bent diagonal and some other features being destroyed. This same trick may be used to make the order-8's magic.

Franklin`s magic circle

Franklin mentioned this(?) magic circle in a letter to Collinson about 1752, although the  circle itself may not have been made public until 1767, as mentioned in the introduction.

• The circle uses the integers from 12 to 75 plus another 12 in the center which is used for all summations.
• The eight numbers in each radius plus the central 12 sum to 360.
• The eight numbers in each circle plus the central 12 sum to 360.
• The eight numbers in each eccentric circle plus the central 12 sum to 360. Franklin’s circle apparently shows 20 of these eccentric circles, but is very hard to read. I show only 8 such circles but obviously there are many more!
• Any half circle in the top or bottom, plus half of the center number sum to 180.
• Any half radius plus half of the center number sum to 180.
• Any four adjacent numbers in an ‘almost square’ plus half of the center number sum to 180. For example, 73 + 14 + 15 + 72 + 6 = 180.
• What other combinations are there?

This square was found on a piece of scrap paper with Franklin’s notes, and had no accompanying explanation. Pasles believes it formed the basis for the design of the above magic circle. He provides a plausible algorithm in his paper for its use in this regard. 17 47 30 36 21 43 26 40 32 34 19 45 28 38 23 41 33 31 46 20 37 27 42 24 48 18 35 29 44 22 39 25 49 15 62 4 53 11 58 8 64 2 51 13 60 6 55 9 1 63 14 52 5 59 10 56 16 50 3 61 12 54 7 57 This order-8 has been previously published only once, as a footnote to the Papers of Benjamin Franklin, 1961. It has most of the same characteristics as the 8x8 shown above (the 2 'B' patterns are not valid).

The reason for this magic circle starting at 12 and having a constant 12 in the center is believed to be so the sum would be 360, signifying the number of degrees in a circle.
This diagram is a modern rendering of Franklin's design. He apparently had 20 eccentric circles in his version. I have limited these to 8 in the interest of improved readability.

Franklin`s other squares

Order-4

This square is semi-pandiagonal associative magic, but is rather a disappointment because it contains nothing new. In fact, it was discovered by Bernard Frénicle de Bessy, before his death in 1675, and was published with his list of all 880 order-4 magic squares in 1693. It is one of the 48 semi-pandiagonal, associative magic squares of Group III, so does not even have the bent-diagonal feature. Franklin was probably unaware of the published list of order-4 squares, and was obviously unaware that there did exist 48 fully magic order-4 bent-diagonal magic squares (group II).

Actually, the order-4 square is an exact duplicate of a much older square. That square appeared in a copper engraving made by Albrecht Durer of Germany in 1514. The two numbers in the middle of the bottom row depict this date.

See my Order-4 page for more information on these squares.
See my Still More Squares page for information on the Durer square.

Order-8

A relatively unknown order-8 was found on a scrap of paper with Franklin’s notes. Because it may have figured in the construction of his magic circle, it is displayed in that section.

Franklin`s new order-16

I have chosen not to show this remarkable square, which has many features not found in his well known square.
Instead I have shown my variation, which was constructed by simply moving 4 columns from the left to the right side.

Continuous nature of Franklin squares

Franklin squares are not fully magic (except for the newly discovered order-16. However, they are continuous in the sense that all patterns that run over an edge continue on the opposite edge as pandiagonal magic squares do.
The 8x8 and 16x16 squares may be transformed to a different bent-diagonal square by moving two rows or columns to the opposite edge with many features being retained. However, because some patterns of these squares start only on every forth row and/or column, 4 rows or columns must be moved to retain all the features of the square.

Franklin’s recently discovered square, unlike the well-known 16 x 16, is fully pandiagonal magic. As mentioned previously, all his squares are continuous because all patterns running off of an edge continue on the opposite edge. However, this square differs because it has a pattern of n adjacent numbers in a straight diagonal. The other squares have only diagonals of n/2 (the reason they are not true magic squares). Of course, it still has the bent-diagonal feature as well!

This variation of Franklin’s newly discovered 16 x 16 magic square was constructed by simply moving four columns from the left side of his square to the right side. Every pattern tested on this square  was also tested on an exact copy of his square (as presented in Pasles paper), to confirm that the features are identical.

Feature comparison    Between Franklin’s Old and New Order-16 squares

• The square illustrated above has the exact same features as Franklin’s new square.

• I will present a condensed comparison list, then sub-sections showing the small patterns, large patterns, bent-diagonal and knight-move patterns tested.

• The word ‘All’ in the following table indicates that the pattern sums correctly if started in ANY of the 256 cells of the square!

• Small patterns: 16 cells  that all fit within one quadrant.

• Large patterns: 4 groups of 4 cells spread over all 4 quadrants.

• I arbitrarily defined the starting cell of the pattern to be the top cell in the leftmost column of the pattern.

Because the new 16 x 16 is pandiagonal magic and all 2 x 2 blocks sum to n/4, it may be thought that this is a Most-perfect magic square. Alas! Cells spaced n/2 along the diagonals do not sum to n+1!
When summing a pattern of a row of 2 cells, within the same row (of 16 cells), there are only two different sums and they alternate. Likewise, when summing a column of 2 adjacent cells, the column of 16 cells will contain only two different (alternating) sums.

Small patterns

Here are the 12 small patterns mentioned in the comparison table above. Each pattern consists of 16 cells arranged within one quadrant. Each pattern shown may be started in any of the 256 cells of either Franklin 16 x 16 square because of the continuous nature (wrap-around) of these squares.
Furthermore, because both squares may be altered by moving 4 rows or columns from 1 side to the other, these patterns also appear 256 times in my version of the 16 x 16 square.

Every pattern I tested that was diagonally and orthogonally symmetrical, summed correctly, so presumably there are many more of this type then the 11 shown here. The only pattern I found that was not diagonally symmetrical but did sum correctly is shown as sp9.

Large patterns

This image shows 12 large patterns of 16 cells. Each pattern consists of 4 cells per quadrant with these cells placed symmetrical to the diagonal.
The image looks complicated but simply focus on one number/color at a time.

Every pattern of this type that I checked summed correctly to 2056 for all 3 order-16 squares so presumably there are a great many more patterns possible. Because of wrap-around, the pattern may be started in any of the 256 cells of the square. For the original square, no 1/4 pattern summed to 1/4 S and no two 1/4 patterns summed to 1/2 S. When these patterns were tested on the new square and my version of it, results for several patterns were different, depending on which of the cells the pattern was started on. In some starting positions for patterns 5, 6 and 7, each of the four 1/4 patterns sum correctly to 1/4 S or the two pairs of 1/4 patterns each sum to 1/2 S. This is explained by the fact that these 1/4 patterns are 4 cell diagonals (see 3 and 4 in the comparison table). I did not find similar characteristics for any of the other nine patterns but did not test all patterns in all starting positions, so there is a possibility that some may exist.

Two midsize patterns tested (MP1 & MP2) are 16 symmetrical cells within a 12x12 and a 14x14 square.
I believe that ANY pattern of 16 cells that are fully symmetrical within a square area from 6x6 to 16 x 16, will sum correctly when the pattern is started in ANY of the 256 cells of the Franklin square.
There is a magic square with similar features, but which includes 4 irregular patterns, on my unusual squares page.

Bent diagonal patterns

Shown here are 9 of the 16 zig-zag patterns tested. The other 7 patterns tested are reflections of these (reflections of 2 were not tested).

My test spreadsheets showed a number in each of the 64 cells of the square. This number was the total for the 16 cells whose pattern started on that cell. These patterns only sum correctly if started on the odd columns, so these columns all contain 2056. However, in most cases, the EVEN columns showed the incorrect totals as two alternating numbers.

Conclusion

I show 12 small patterns (SP), 12 large patterns (LP), 16 bent diagonals (ZZ), and 4 knight move patterns (KN). All these patterns appear in all positions or many ordered positions in one or both of Franklin’s 16 x 16 squares as detailed in the comparison table

Be aware that there are many more such patterns. Maybe you will choose to investigate these fascinating squares further. If so, I would appreciate being advised of additional patterns that you find!

Order-8 Franklin squares counted!

Recently Daniel Schindel,Matthew Rempel And Peter Loly (Winnipeg, Canada) counted the basic Franklin type bent-diagonal squares of order-8. [1]

There are exactly 1,105,920 of them. Two-thirds of these squares are not magic because the main diagonals do not sum correctly. Exactly one-third (368,640) are pandiagonal magic.
BTW The Peter Loly's count has been independently corroborated by other sources in Canada and Argentina.
An interesting report of this event appeared in Ivars Peterson's Mathtrek column in Science News Online (June 24, 2006) [2]

This figure (368,640) is in exact agreement with that reported by Dame Kathleen Ollerenshaw as being most-perfect. The bent-diagonal pandiagonal squares all have the 2z2 feature (compact), but fail on the diagonal feature (complete) so we can assume that there are no order-8 bent-diagonal most-perfect magic squares!

Review of requirements to be classed as most-perfect: [3]

1.      Doubly-even pandiagonal normal magic squares (i.e. order 4, 8, 12, etc using integers from 1 to m²)

2.      Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= m² + 1) (compact)

3.      Any pair of integers distant ½m along a diagonal sum to T (complete)

Compact magic squares

All Franklin magic squares with correct main diagonals are pandiagonal magic. They all have the compact feature (all 2x2 blocks of cells sum to 4/m of S).
I have recently (May 2007) added a page explaining new findings of compact magic squares. As examples, I compare 4 Franklin type order 8 squares. [4]
I have an Excel spreadsheet used in conjunction with this page. It is available for downloading. [5]

[1] Proc. R. Soc. A (2006) 462, 2271–2279, doi:10.1098/rspa.2006.1684. Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS
Published online 28 February 2006. Obtainable by download from Peter Loly's home page
[2] http://www.sciencenews.org/articles/20060624/ No longer available?
[3] My Most-perfect page.
[4] My Compact magic squares page
[5] Compact_8-MS.xls on my Downloads page

	This page was originally posted September 2001 It was last updated February 11, 2013 Harvey Heinz   harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz