Constructing Franklin Magic Squares

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1 Constructing Franklin Magic Squares Cor Hurkens Mathematics Department TU Eindhoven Kubusdag Eindhoven 14 oktober 2007 impossible number puzzles / department of mathematics and computer science 1/21

2 Benjamin Franklin discovered squares with numbers 1,..., 64 arranged / department of mathematics and computer science 2/21

3 Benjamin Franklin discovered squares with numbers 1,..., 64 arranged each ROW having sum = 260. / department of mathematics and computer science 2/21

4 Benjamin Franklin discovered squares with numbers 1,..., 64 arranged each ROW having sum = 260, each COLUMN having sum = 260. / department of mathematics and computer science 2/21

5 Benjamin Franklins square is NOT magic DIAGONALS have sums 262 and 258 / department of mathematics and computer science 3/21

6 Benjamin Franklins square is NOT magic but BENT DIAGONALS have sums 260. / department of mathematics and computer science 3/21

7 Benjamin Franklins square is NOT magic but BENT DIAGONALS have sums 260 (in both orientations). / department of mathematics and computer science 3/21

8 Benjamin Franklin squares are more than magic: SHIFTED BENT DIAGONALS have magic sum 260 (all types, >,, <) / department of mathematics and computer science 4/21

9 Franklin Magic squares further feature HALF ROWS having fixed sum ( = 130) / department of mathematics and computer science 5/21

10 Franklin Magic squares further feature HALF ROWS having fixed sum ( = 130), HALF COLUMNS having fixed sum ( = 130) / department of mathematics and computer science 5/21

11 Franklin Magic squares finally have x2 SUBSQUARES all have sum = 130 even if wrapped-around / department of mathematics and computer science 6/21

12 a general n by n Franklin Magic square M has 1. entries 1, 2,..., n 2 ; 2. each row and each column has a fixed entry sum n(1 + n 2 )/2; [ ] Mi,j M 3. each two by two sub-square i,j+1 has sum 2(1 + n M i+1,j M 2 ); i+1,j+1 4. each half row starting in column 1 or n/2 + 1 has sum of entries equal to n(1+n 2 )/4, and similar for half columns starting in row 1 or n/2+1; 5. each shifted bent diagonal has entry sum equal to n(1 + n 2 )/2. / department of mathematics and computer science 7/21

13 a general n by n Franklin Magic square M has 1. entries 1, 2,..., n 2 ; 2. each row and each column has a fixed entry sum n(1 + n 2 )/2; [ ] Mi,j M 3. each two by two sub-square i,j+1 has sum 2(1 + n M i+1,j M 2 ); i+1,j+1 4. each half row starting in column 1 or n/2 + 1 has sum of entries equal to n(1+n 2 )/4, and similar for half columns starting in row 1 or n/2+1; 5. each shifted bent diagonal has entry sum equal to n(1 + n 2 )/2. CHALLENGE: find more Franklin Magic squares! REQUIREMENT: n is multiple of 4 / department of mathematics and computer science 7/21

14 a 4 by 4 Franklin Magic square would have entries 1,..., 16 X??????????????? / department of mathematics and computer science 8/21

15 a 4 by 4 Franklin Magic square would have entries 1,..., 16 X 17 X?? 17 X??????????? half rows, half columns have sum 17 / department of mathematics and computer science 8/21

16 a 4 by 4 Franklin Magic square would have entries 1,..., 16 X 17 X?? 17 X X?????????? half rows, half columns have sum 17 2x2 subsquares have sum 34 / department of mathematics and computer science 8/21

17 a 4 by 4 Franklin Magic square would have entries 1,..., 16 X 17 X?? 17 X X?????????? half rows, half columns have sum 17 2x2 subsquares have sum 34 but entries should be distinct! No such thing! / department of mathematics and computer science 8/21

18 a 16 by 16 Franklin Magic square must have entries 1,..., / department of mathematics and computer science 9/21

19 Some facts a 4 by 4 Franklin Magic square does not exist Franklin produced some 8 by 8 and some 16 by 16 Franklin squares (±1750) web sites of magic squares display Franklin Magic squares of order 8 and 16, even 32 constructing squares of order 8k is not so difficult March 22, 2007: HYPE in the Netherlands high school students Hoekstra, Schulte, Alkema construct an ALMOST- FRANKLIN-MAGIC square of order 12 (Dec 14, 2006) Question do there exist Franklin Magic squares of order 12, 20, 28,...? / department of mathematics and computer science 10/21

20 HSA Magic Square / department of mathematics and computer science 11/21

21 HSA Magic Square by 2 squares have sum 4 1 ( ) = / department of mathematics and computer science 11/21

22 HSA Magic Square by 2 squares have sum 4 1 ( ) = each third row or column has sum 290 / department of mathematics and computer science 11/21

23 HSA Magic Square by 2 squares have sum 4 1 ( ) = each third row or column has sum 290 each bent-diagonal has sum 12 1 ( ) = / department of mathematics and computer science 11/21

24 Questions do there exist Franklin Magic squares of order 12, 20, 28,...? can a brute-force search by computer find such square? or is there some sophisticated way to construct these squares? Answers: Yes and No / department of mathematics and computer science 11/21

25 Searching for a Franklin magic square of order 12 / department of mathematics and computer science 12/21

26 Searching for a Franklin magic square of order 12 RED equals BLUE equals GREEN. / department of mathematics and computer science 12/21

27 Searching for a Franklin magic square of order 12 RED equals BLUE equals GREEN. RED plus YELLOW is magic, hence YELLOW plus GREEN is magic. / department of mathematics and computer science 12/21

28 Searching for a Franklin magic square of order 12 / department of mathematics and computer science 13/21

29 Searching for a Franklin magic square of order 12 YELLOW plus GREEN is magic; GREEN plus BLUE is magic / department of mathematics and computer science 13/21

30 Searching for a Franklin magic square of order 12 YELLOW plus GREEN is magic; GREEN plus BLUE is magic YELLOW equals BLUE! / department of mathematics and computer science 13/21

31 Searching for a Franklin magic square of order 12 Bent-diagonal with magic sum / department of mathematics and computer science 14/21

32 Searching for a Franklin magic square of order 12 Sum remains magic / department of mathematics and computer science 14/21

33 Searching for a Franklin magic square of order 12 Sum remains magic / department of mathematics and computer science 14/21

34 Searching for a Franklin magic square of order 12 Sum remains magic / department of mathematics and computer science 14/21

35 Searching for a Franklin magic square of order 12 Sum remains magic / department of mathematics and computer science 14/21

36 Searching for a Franklin magic square of order 12 Sum remains magic / department of mathematics and computer science 14/21

37 Searching for a Franklin magic square of order 12 Sum remains magic / department of mathematics and computer science 14/21

38 Searching for a Franklin magic square of order 12 magic RED sum equals sum of THREE magic 2x2 squares / department of mathematics and computer science 15/21

39 Searching for a Franklin magic square of order 12 RED equals YELLOW (only used 2x2 magic subsquare property) / department of mathematics and computer science 15/21

40 Searching for a Franklin magic square of order 12 RED equals YELLOW (only used 2x2 magic subsquare property) RED plus YELLOW is magic ROW (870) / department of mathematics and computer science 15/21

41 Searching for a Franklin magic square of order 12 RED has magic sum (435) used 2x2 magic subsquare property, plus magic row sum / department of mathematics and computer science 15/21

42 Let P = {1, 3, 5}, Q = {2, 4, 6}, R = {7, 9, 11}, S = {8, 10, 12}, a Franklin Magic square M of order 12 is a square with 1. entries 1, 2,..., 144; 2. each two by two sub-square [ ] Mi,j M i,j+1 has sum 290; M i+1,j M i+1,j+1 3. in row (column) 1, entries in first half (P + Q) add up to 435; 4. in row (column) 1, entries in second half (R + S) add up to 435; 5. in row (column) 1, entries in first half odd positions plus entries in second half of even positions (P + S) add up to 435; Note: in row 1, entries in P and entries in R have the same sum / department of mathematics and computer science 16/21

43 Let P = {1, 3, 5}, Q = {2, 4, 6}, R = {7, 9, 11}, S = {8, 10, 12}, a Franklin Magic square M of order 12 remains Franklin Magic even 1. reflected along diagonal 2. reflected along horizontal axis of symmetry 3. reflected along vertical axis of symmetry 4. interchanging columns 1, 3, 5 (similar for Q, R, S); 5. interchanging columns in P with those in R (similar for Q and S); 6. applying above tricks to rows / department of mathematics and computer science 17/21

44 Position of 1 in a Franklin magic square of order 12 x x x x x x x x 1 x x x / department of mathematics and computer science 18/21

45 Position of 1 in a Franklin magic square of order 12 x x 1 x x x x x x x x x / department of mathematics and computer science 18/21

46 Position of 1 in a Franklin magic square of order 12 x x 1 x x x x x x x x x / department of mathematics and computer science 18/21

47 Position of 1 in a Franklin magic square of order 12 1 x x x x x x x x x x x / department of mathematics and computer science 18/21

48 Position of 1 in a Franklin magic square of order 12 1 x x x x x x x x x x x / department of mathematics and computer science 18/21

49 Position of 1 in a Franklin magic square of order 12 1 x x x x x x x x x x x / department of mathematics and computer science 18/21

50 Position of 1 in a Franklin magic square of order 12 1 x x x x x x x x x x x if there is a 12 by 12 FMS, there is one starting with 1 / department of mathematics and computer science 18/21

51 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x Fill in first row and column, rest follows from 2x2 squares / department of mathematics and computer science 19/21

52 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x Fill in first row and column, rest follows from 2x2 squares M 22 = x 1 y 1 ; M 23 = 290 x 1 x 2 M 22 / department of mathematics and computer science 19/21

53 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x First row: x 5 (in Q), x 10 (in R), x 11 (in S) follow from rules Similar: y 5, y 10, and y 11 are fixed; 16 degrees of freedom. / department of mathematics and computer science 20/21

54 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x rough number of possibilities: / department of mathematics and computer science 20/21

55 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x number of possibilities: / department of mathematics and computer science 21/21

56 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x number of possibilities: / department of mathematics and computer science 21/21

57 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x number of possibilities: / department of mathematics and computer science 21/21

58 Searching for a Franklin magic square of order 12 1 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 y 1 x x x x x x x x x x x y 2 x x x x x x x x x x x y 3 x x x x x x x x x x x y 4 x x x x x x x x x x x y 5 x x x x x x x x x x x y 6 x x x x x x x x x x x y 7 x x x x x x x x x x x y 8 x x x x x x x x x x x y 9 x x x x x x x x x x x y 10 x x x x x x x x x x x y 11 x x x x x x x x x x x number of possibilities: , NONE of which extend to 12 by 12! 165 hours of computing time (3.5 hours using 50 computers in parallel). / department of mathematics and computer science 21/21

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