

Contents Introduction A bit about Ben Franklin. Then what this page is about and a credit to Paul C. Pasles. Franklin`s Order-8 This well-known square was mentioned in a letter about 1752 and first published in 1769. The well known 16 x 16 This well-known 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 order-4, order-6, and preliminary mention of the remarkable order-16. Franklin`s new order-16 A variation of the newly discovered square and a comparison with the well-known one. Feature comparison Feature comparison table between Franklin’s old and new order-16 squares. Small patterns An arbitrary selection of 12 patterns of 16 cells found in the order-16 squares. Large patterns An arbitrary selection of 12 patterns of 16 cells found in the order-16 squares. Bent diagonal patterns An arbitrary selection of 16 zig-zag patterns of 16 cells found in the order-16 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. Order-8 Franklin counted! There are 368,640 order-8 bent-diagonal pandiagonal magic squares. (Peter Loly 2006.) 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. 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 order-4, an order-6, another order-8 and another order-16.

This latter square has the usual bent-diagonals but in addition is a pandiagonal magic square! 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 52 61 4 13 20 29 36 45 14 3 62 51 46 35 30 19 53 60 5 12 21 28 37 44 11 6 59 54 43 38 27 22 55 58 7 10 23 26 39 42 9 8 57 56 41 40 25 24 50 63 2 15 18 31 34 47 16 1 64 49 48 33 32 17 This square has Franklin's trademark, the bent-diagonals. 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 order-8 and order-16 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 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. 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`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. 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 2 9 4 29 36 31 34 32 30 7 5 3 6 1 8 33 28 35 20 27 22 11 18 13 25 23 21 16 14 12 24 19 26 15 10 17 Order-6 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 order-3 squares. By simply swapping the two halves of the second row from the top, these top two quadrants also become order-3 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. 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! 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. 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. # 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 order-4 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 semi-magic 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, 10-12 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 4-cell, start down None All starting on Any ODD row & column 39 ZZ4 – horizontal 8 cell segment, then two 4-cell, 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 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. 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 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. # 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 tested 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 order-16 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 Order-8

Order-16

Magic Circle Both squares feature his famous ‘bent-diagonal’s’ but neither is magic in the accepted sense. All these were published in his lifetime, and many times since. Order-8 This is also bent-diagonal but not magic and was first published in 1959. Order-4 Order-6 Order-16 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, June-July, 2001, pp 489-511. The order-4 is actually a Disguised version of Frénicle’s # 175 magic square. It is associated but does not have bent diagonals. The order-6 is not magic but has bent diagonals. The order-16 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? 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