Introduction 
A bit about Ben Franklin. Then what this page is about and a credit to Paul C. Pasles. 
Franklin’s Order8 
This wellknown square was mentioned in a letter about 1752 and first published in 1769. 
The well known 16x16 
This wellknown square was mentioned in a letter about 1752 and first published in 1767. 
Franklin’s magic circle 
This circle was also published in Franklin's lifetime and is fairly well known. 
Franklin’s Other squares 
The newly discovered order4, order6, and preliminary mention of the remarkable order16. 
Franklin's new order16 
A variation of the newly discovered square and a comparison with the wellknown one. 
Feature Comparison 
Feature comparison table between Franklin’s old and new order16 squares. 
Small patterns 
An arbitrary selection of 12 patterns of 16 cells found in the order16 squares. 
Large patterns 
An arbitrary selection of 12 patterns of 16 cells found in the order16 squares. 
Bent diagonal patterns 
An arbitrary selection of 16 zigzag patterns of 16 cells found in the order16 squares. 
Knight move diagonals 
Diagrammed are 4 knight move diagonal patterns found in both squares. 
Conclusion 
A summery of the information contained on this page. 
Order8 Franklin Squares Counted! 
There are 368,640 order8 bentdiagonal pandiagonal magic squares. Peter Loly 2006. 
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. 
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/franklinsquares.html Oops! gone now?

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. 

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 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.
200  217  232  249  8  25  40  57  72  89  104  121  136  153  168  185 
58  39  26  7  250  231  218  199  186  167  154  135  122  103  90  71 
198  219  230  251  6  27  38  59  70  91  102  123  134  155  166  187 
60  37  28  5  252  229  220  197  188  165  156  133  124  101  92  69 
201  216  233  248  9  24  41  56  73  88  105  120  137  152  169  184 
55  42  23  10  247  234  215  202  183  170  151  138  119  106  87  74 
203  214  235  246  11  22  43  54  75  86  107  118  139  150  171  182 
53  44  21  12  245  236  213  204  181  172  149  140  117  108  85  76 
205  212  237  244  13  20  45  52  77  84  109  116  141  148  173  180 
51  46  19  14  243  238  211  206  179  174  147  142  115  110  83  78 
207  210  239  242  15  18  47  50  79  82  111  114  143  146  175  178 
49  48  17  16  241  240  209  208  177  176  145  144  113  112  81  80 
196  221  228  253  4  29  36  61  68  93  100  125  132  157  164  189 
62  35  30  3  254  227  222  195  190  163  158  131  126  99  94  67 
194  223  226  255  2  31  34  63  66  95  98  127  130  159  162  191 
64  33  32  1  256  225  224  193  192  161  160  129  128  97  96  65 
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.
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.
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.
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, probably around 1665, and was published with his list of all 880
order4 magic squares in 1676. 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).
See my Order4 page for more information on these squares.


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.
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!
14  253  4  243  12  251  6  245  10  249  8  247  16  255  2  241 
3  244  13  254  5  246  11  252  7  248  9  250  1  242  15  256 
238  29  228  19  236  27  230  21  234  25  232  23  240  31  226  17 
227  20  237  30  229  22  235  28  231  24  233  26  225  18  239  32 
221  46  211  36  219  44  213  38  217  42  215  40  223  48  209  34 
212  35  222  45  214  37  220  43  216  39  218  41  210  33  224  47 
61  206  51  196  59  204  53  198  57  202  55  200  63  208  49  194 
52  195  62  205  54  197  60  203  56  199  58  201  50  193  64  207 
78  189  68  179  76  187  70  181  74  185  72  183  80  191  66  177 
67  180  77  190  69  182  75  188  71  184  73  186  65  178  79  192 
174  93  164  83  172  91  166  85  170  89  168  87  176  95  162  81 
163  84  173  94  165  86  171  92  167  88  169  90  161  82  175  96 
157  110  147  100  155  108  149  102  153  106  151  104  159  112  145  98 
148  99  158  109  150  101  156  107  152  103  154  105  146  97  160  111 
125  142  115  132  123  140  117  134  121  138  119  136  127  144  113  130 
116  131  126  141  118  133  124  139  120  135  122  137  114  129  128  143 
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.
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 !!!

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 
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.
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!
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.
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 
Knight move diagonals testedI 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. 
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 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. 
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.htm
Please send me Feedback about my Web site!
Last updated
September 12, 2009
Harvey Heinz harveyheinz@shaw.ca
Copyright © 2001 by Harvey D. Heinz