Contents

A bit about Ben
Franklin. Then what this page is about and a credit to Paul C.
Pasles. 

This wellknown square
was mentioned in a letter about 1752 and first published in 1769. 

This wellknown square
was mentioned in a letter about 1752 and first published in 1767. 

This circle was also
published in Franklin's lifetime and is fairly well known. 

The newly discovered
order4, order6, and preliminary mention of the remarkable
order16. 

A variation of the
newly discovered square and a comparison with the wellknown one. 

Feature
comparison table between Franklin’s old and new order16 squares. 

An arbitrary selection
of 12 patterns of 16 cells found in the order16 squares. 

An arbitrary selection
of 12 patterns of 16 cells found in the order16 squares. 

An arbitrary selection
of 16 zigzag patterns of 16 cells found in the order16 squares. 

Diagrammed are 4 knight
move diagonal patterns found in both squares. 

A summery of the
information contained on this page. 

There are 368,640
order8 bentdiagonal pandiagonal magic squares. (Peter Loly 2006.) 
Introduction
Benjamin Franklin (17061790), the early American
scientist, statesman and author, is known as the creator of bentdiagonal
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 order8, an order16 and a
magic circle. The order16 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.
It
has recently been revealed in a superbly researched and written paper by
Paul C. Pasles that, in fact, Franklin composed four other squares; an
order4, an order6, another order8 and another order16.
This latter square has the usual bentdiagonals but in addition is a
pandiagonal magic square! 
The orders 4, 6 and 8 will be reproduced below. A variation
of the order16 (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, JuneJuly, 2001, pp 489511. 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 Order8
square
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This square has Franklin's trademark, the bentdiagonals.
However, because the main diagonals do not sum correctly (one totals 260 
32 & the other 260 + 32), it is not a true magic square. The Franklin
order8 and order16 squares have the feature that you can move a group of 4
rows or 4 columns to the opposite side and all features will be retained.
This magic square was constructed by Benjamin Franklin
and first mentioned in a letter to Peter Collinson of England about 1752. It
was subsequently published in Franklin's Experiments and Observations in
Electricity (London, 1569)
It has many interesting properties as illustrated by
the following cell patterns. 

 Because of wraparound, 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 order8 square,
plus many more. It contains the bent diagonals which are a trademark of
his squares. However, like his order8, 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
order8's magic.
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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.
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This order8 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
Order4
This square is semipandiagonal 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 order4 magic squares in 1693. It is one of the 48
semipandiagonal, associative magic squares of Group III, so does not even have
the bentdiagonal feature. Franklin was probably unaware of the published list
of order4 squares, and was obviously unaware that there did exist 48 fully
magic order4 bentdiagonal magic squares (group II).
Actually, the order4 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 Order4 page for more information on these
squares.
See my Still
More Squares page for information on the Durer square.
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Order6
The main diagonals are bent, i.e 2+32+8+33+5+31=111
so this square is not a true magic square. However, note that the two
bottom quadrants are magic order3 squares. By simply swapping the two
halves of the second row from the top, these top two quadrants also become
order3 magic. If this 6 x 6 square is split into two horizontal 3 x 6
rectangles, each rectangle is associated. That is, diametrically opposed
numbers such as 2 + 35, 32 + 5, 27+10, etc. sum to 37.
Neither the 4 x 4 or the 6 x 6 square had been
published previous to Pasles paper. 
Order8
A relatively unknown order8 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
order16
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 order16. 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 bentdiagonal 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
wellknown 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 bentdiagonal feature as well!
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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
Order16 squares

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

I will present a condensed
comparison list, then subsections showing the small patterns, large
patterns, bentdiagonal and knightmove 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.
# 
Pattern Description 
Original 16 x 16 
New 16 x 16 (& modified one) 
1 
4 x 4 blocks = S 
All 
All 
2 
2 x 2 blocks = S/4 
All 
All 
3 
Leading 4 cell diagonals = S/4 
None 
Start on Any ODD column & rows 3, 7, 11, 15

4 
Right 4 cell diagonals = S/4 
None 
Start on Any ODD column & rows 2, 6, 10, 14 
5 
Leading 8 cell diagonals = S/2 
None 
Start on ANY ODD row & col. 
6 
Right 8 cell diagonals = S/2 
None 
Start on ANY even row & ODD column 
7 
Leading 16 cell diagonals = S 
None 
All  This (and next) is what makes this square magic
(and pandiagonal)! 
8 
Right 16 cell diagonals = S 
None 
All 
9 
Rows of 2 cells = S/8 
All starting with col. 4 & 12 
All starting with columns 2, 6, 10, 14 
10 
Columns of 2 cells = S/8 
None ! 
None !!! 
11 
Any row of 4 cells = S/4 
All starting on columns 3 & 11 
All starting on col. 1, 5, 9, 13 
12 
Any column of 4 cells = S/4 
All starting on rows 3 & 11 
NO columns of 4 cells sum to S/4 !!!
(so there can be NO embedded order4 magic squares) 
13 
Any row of 8 cells = S/2 
All starting on columns 1 & 9 
All starting on col. 1, 5, 9, 13 
14 
Any column of 8 cells = S/2 
All starting on rows 1 & 9 
All starting on rows 1, 5, 9, 13 
15 
Is this square magic? 
No 
Yes – pandiagonal magic! 
16 
Embedded 8 x 8 magic squares
(No 4 x 4 are magic) 
The one in each quadrant is semimagic 
All starting on rows & columns 1, 5 , 9, 13 are
pandiagonal magic. 
17 
Corners of 4 x 4 
All 
All 
18 
Corners of 6 x 6 
All 
All 
19 
Corners of 8 x 8 
All 
All 
20 
Corners of 10 x 10 
All 
All 
21 
Corners of 12 x 12 
All 
All 
22 
Corners of 14 x 14 
All 
All 
23 
Corners of 16 x 16 
All 
All 
24 
Small Pattern 1 – diamond 
All 
All 
25 
SP2 – box 
All 
All 
26 
SP3 – large x 
All 
All 
27 
SP4 – small diamond 
All 
All 
28 
SP5 to 8, 1012 
All 
All 
35 
Pattern 9 – NOTE this is the only pattern found so
far that is NOT diagonally symmetrical (but works 
All 
All 
36 
ZZ1 – horizontal 4 cell segments, start down 
All starting on ANY ODD column 
All starting on ANY ODD column 
37 
ZZ2 – horizontal 4 cell segments, start up 
All starting on ANY ODD column 
All starting on ANY ODD column 
38 
ZZ3 – horizontal 8 cell segment, then two 4cell,
start down 
None 
All starting on Any ODD row & column 
39 
ZZ4 – horizontal 8 cell segment, then two 4cell,
start up 
None 
All starting on Any EVEN row & ODD column 
40 
ZZ5 – Vertical 4 cell segments, start right 
All starting on rows 3, 7, 11, 15 
All starting on rows 3, 7, 11, 15 
41 
ZZ6 – Vertical 4 cell segments, start left 
All starting on rows 2, 6, 10,, 14 
All starting on rows 2, 6, 10, 14 
42 
ZZ7 – Horizontal, 2 cell segments, start down 
All starting on ODD columns 
All starting on ODD columns 
43 
ZZ8 – Horizontal, 2 cell segments, start up 
All starting on ODD columns 
All starting on ODD columns 
44 
ZZ9 – Vertical, 2 cell segments, start right 
All starting on ODD rows 
All starting on ODD rows 
45 
ZZ10 – Vertical, 2 cell segments, start left 
All starting on EVEN rows 
All starting on EVEN rows 
46 
ZZ11 – Horizontal, 6, 4, 2, 2 segments 
None 
Start on rows 3, 7, 11, 15 and ODD columns 
47 
ZZ12 – Vertical, 6, 4, 2, 2 segments 
None 
Only some 
48 
ZZ13 – 8 cell segments, horizontal, down/up 
All starting on ODD columns 
All starting on ODD columns 
49 
ZZ14 – 8 cell segments, horizontal, up/down 
All starting on ODD columns 
All starting on ODD columns 
50 
ZZ15 – 8 cell segments, vertical, right/left 
All starting only on rows 1 & 9 
All starting on ODD rows 
51 
ZZ16 – 8 cell segments, vertical, left/right 
All starting only on rows 8 & 16 
All starting on EVEN rows 
52 
Knight moves Diagonal, vertical right
KM1 
All 
All 
53 
Knight moves Diagonal, vertical left KM2 
All 
All 
54 
Knight moves Diagonal, horizontal, down
KM3 
All 
All 
55 
Knight moves Diagonal, horizontal, up
KM4 
All 
All 
56 
LP1 – Large pattern # 1 
All 
All 
57 
LP2 – LP12 
All 
All 
68 
MP1 & MP2 
All 
All 
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 Mostperfect 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
(wraparound) 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.
All 12 patterns shown here sum correctly when started in any of the 256 cells
of the original Franklin square, his newly discovered one, and my version of
that one!
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 order16 squares so presumably there are a great
many more patterns possible. Because of wraparound, 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 zigzag 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.
# 
Original square 
New square 
ZZ1 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 2048 and
2064 
ZZ2 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 2048 and
2064 
ZZ7 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 1992 and
2120 
ZZ8 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 1992 and
2120 
ZZ9 

The EVEN rows have alternating sums of 1800 and 2312 
ZZ10 

The ODD rows have alternating sums of 1800 and 2312 
ZZ13 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 2048 and
2064 
ZZ14 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 2048 and
2064 
ZZ15 
The EVEN columns have alternating sums of 1800 and
2312 
The EVEN columns have alternating sums of 2048 and
2064 
ZZ16 

The ODD rows have alternating sums of 1928 and 2184 
I tested 9 knight move patterns and found only 4
that sum correctly. However, undoubtedly there are more valid ones as
yet undiscovered.
Knight move 2 is a horizontal reflection of 1 and
knight move 3 is a vertical reflection of 4. Keep in mind that the
patterns shown in these diagrams do not represent the placement in the
order16 square. Each pattern may appear in that square in any
position, subject to the conditions mentioned in the comparison
summary table. 

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!
Franklin’s Squares 
Order8
Order16
Magic Circle 
Both squares feature his famous ‘bentdiagonal’s’ but
neither is magic in the accepted sense. All these were published in his
lifetime, and many times since. 
Order8 
This is also bentdiagonal but not magic and was first
published in 1959. 
Order4 Order6
Order16 
These three squares have never been published prior
to
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, JuneJuly, 2001, pp 489511.
The order4 is actually a Disguised version of
Frénicle’s # 175 magic square. It is associated but does not have bent
diagonals.
The order6 is not magic but has bent diagonals.
The order16 is a pandiagonal magic square but also
has many versions of bent diagonals. It has many more magic patterns then
the version Franklin published.
Surely this is "the most magically magic of any magic
square".
Why did he never publish it? 
Order8 Franklin
squares counted!
Recently Daniel Schindel,Matthew Rempel
And Peter Loly (Winnipeg, Canada) counted the basic Franklin type
bentdiagonal squares of order8. [1]
There are exactly 1,105,920 of them. Twothirds of these squares are not magic
because the main diagonals do not sum correctly.
Exactly onethird (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 mostperfect. The bentdiagonal pandiagonal squares all
have the 2z2 feature (compact), but fail on the diagonal feature (complete) so
we can assume that there are no order8 bentdiagonal mostperfect magic
squares!
Review of requirements to be classed as mostperfect:
[3]
1.
Doublyeven pandiagonal normal magic squares (i.e. order 4, 8, 12, etc
using integers from 1 to m^{2})
2.
Every 2 x 2 block of cells (including wraparound) sum to 2T (where T=
m^{2} + 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 Mostperfect page.
[4] My Compact magic squares page
[5] Compact_8MS.xls on my Downloads
page
