Magic Squares - University of Utah

1 Magic SquaresJenny KenkelSeptember 4, 2018 DefinitionAmagic squareis an ngrid of numbers such that the sum ofeach row is equal, and equal to the sum of each definitions also require the sum along the main diagonals toadd to the same square is an nsquare in which each of theentries 1,..,n2is used exactly once, and one in which the sum ofthe main diagonals is equal to the row (and column) Squares : HistoryIThere is a legend that the (semi-mythical) emperor Yu, BCE, copied a Magic square off the back of a giantturtle in the Luo, a tributary of the Huang He (Yellow River).

2 IThe turtle s Magic square is called theLuo Shuand is492357816 IThis story originated no later than 200 Hui s ConstructionsThe following method for constructing the Luo Shu andconstructing a 4 4 Magic square come from Yang Hui s book,1275 CE: Xu Gu Zhai Suan Fa Continuation of Ancient Mathematical Methods for Elucidatingthe Strange Properties of Numbers Method for Constructing the Luo Shu (c. 1275)IArrange numbers so thatthey slant downward, to therightIInterchange the top and thebottom (1 and 9)IInterchange the left andrightmost entries (7 and 3)ILower 9 to fill the slotbetween 4 and 2, raise 2 tofill the slot between 8 and 6I142753869I942357861I492357816 Method for Constructing the Luo Shu (c.)

3 1275)IArrange numbers so thatthey slant downward, to therightIInterchange the top and thebottom (1 and 9)IInterchange the left andrightmost entries (7 and 3)ILower 9 to fill the slotbetween 4 and 2, raise 2 tofill the slot between 8 and 6I142753869I942357861I492357816 Method for Constructing the Luo Shu (c. 1275)IArrange numbers so thatthey slant downward, to therightIInterchange the top and thebottom (1 and 9)IInterchange the left andrightmost entries (7 and 3)ILower 9 to fill the slotbetween 4 and 2, raise 2 tofill the slot between 8 and 6I142753869I942357861I492357816 Method for Constructing the Luo Shu (c.

4 1275)IArrange numbers so thatthey slant downward, to therightIInterchange the top and thebottom (1 and 9)IInterchange the left andrightmost entries (7 and 3)ILower 9 to fill the slotbetween 4 and 2, raise 2 tofill the slot between 8 and 6I142753869I942357861I492357816 Yang Hui s Method of Constructing 4 4 Magic SquaresIArrange the numbers 1 to16 in order in a 4 4 arrayIInterchange the numbers inthe corner of the outersquareIInterchange the numbers atthe corners of the innersquareI12345678910111213141516I1623 1356789101112414151I16231351110897612414 151 Yang Hui s Method of Constructing 4 4 Magic SquaresIArrange the numbers 1 to16 in order in a 4 4 arrayIInterchange the numbers inthe corner of the outersquareIInterchange

5 The numbers atthe corners of the innersquareI12345678910111213141516I1623 1356789101112414151I16231351110897612414 151 Yang Hui s Method of Constructing 4 4 Magic SquaresIArrange the numbers 1 to16 in order in a 4 4 arrayIInterchange the numbers inthe corner of the outersquareIInterchange the numbers atthe corners of the innersquareI12345678910111213141516I1623 1356789101112414151I16231351110897612414 151 Neat Properties of Yang Hui s 4 4 squareNote that the sum of each quadrant is 34, the same as therow/column sum, as is the sum of the four outer corners, and thecenter Properties of Yang Hui s 4 4 squareNote that the sum of each quadrant is 34, the same as therow/column sum, as is the sum of the four outer corners, and thecenter Properties of Yang Hui s 4 4 squareNote that the sum of each quadrant is 34, the same as therow/column sum, as is the sum of the four outer corners, and thecenter Tangent: Magic Square in Durer s Melancholia(1514)IIncludes a 4 4 magicsquare!

6 IEtched in 1514 Durer s Melancholia (1514) and the DaVinci Code Exactly, Langdon said. But didyou know that this Magic square isfamous because D urer accomplishedthe seemingly impossible? He quicklyshowed Katherine that in addition tomaking the rows, columns, and diag-onals add up to thirty- four , D urer hadalso found a way to make the fourquadrants, the four center Squares ,and even the four corner Squares addup to that number. Most amazing,though, was D urer s ability to posi-tion the numbers 15 and 14 togetherin the bottom row as an indication ofthe year in which he accomplished thisincredible feat!

7 Katherine scanned the numbers,amazed by all the Dan Brown,the DaVinci CodeYang Hui s 7 7 Magic square4681620297493401214184147443733231 9136281511252935225243127172645489383632 102142343021434 Contains a 5 5 Magic square and a 3 3 Magic square!Yang Hui did not write how he found it!Counting Perfect Magic SquaresThere are no perfect 2 2 Magic Squares :1234,1432 Once I ve fixed one entry, I need two entries to be non-distinct:11 FFCounting Perfect Magic SquaresThere are no perfect 2 2 Magic Squares :1234,1432 Once I ve fixed one entry, I need two entries to be non-distinct:11FF3 3 perfect Magic squaresISuppose we want a 3 3 perfect Magic square, , using thedigits 1,2.

8 , + 2 + 3 + + 9 = (9)(10)/2 = 45 Ithus row sum (and column sum) = 45/3 = 15 Inote that whatever number is in the middle needs to be partof 4 different ways to add up to 153 3 perfect Magic squaresISuppose we want a 3 3 perfect Magic square, , using thedigits 1,2,.., + 2 + 3 + + 9 = (9)(10)/2 = 45 Ithus row sum (and column sum) = 45/3 = 15 Inote that whatever number is in the middle needs to be partof 4 different ways to add up to 153 3 perfect Magic squaresISuppose we want a 3 3 perfect Magic square, , using thedigits 1,2.

9 , + 2 + 3 + + 9 = (9)(10)/2 = 45 Ithus row sum (and column sum) = 45/3 = 15 Inote that whatever number is in the middle needs to be partof 4 different ways to add up to 153 3 perfect Magic squaresRow (and column) sum is 15I45 + 3c= 60 (every entry, overcounting the center square 3times, gives me 4 15)3 3 perfect Magic squaresRow (and column) sum is 15I45 + 3c= 60 (every entry, overcounting the center square 3times, gives me 4 15)3 3 Magic squaresIRow (and column) sum is 15 ICenter entry must be 55 Iwhatever numbers go in a diagonal spot must be in 3 differentpartitions of 15I1 can t be in a diagonal:1+(6+8)= 15,1+(5+9)=15I3 can t be in a diagonal:3+(7+5), 3+(8+4)I951I975313 3 Magic squaresIRow (and column) sum is 15 ICenter entry must be 55 Iwhatever numbers go in a diagonal spot must be in 3 differentpartitions of 15I1 can t be in a diagonal:1+(6+8)= 15,1+(5+9)=15I3 can t be in a diagonal.

10 3+(7+5), 3+(8+4)I951I975313 3 Magic squaresIRow (and column) sum is 15 ICenter entry must be 55 Iwhatever numbers go in a diagonal spot must be in 3 differentpartitions of 15I1 can t be in a diagonal:1+(6+8)= 15,1+(5+9)=15I3 can t be in a diagonal:3+(7+5), 3+(8+4)I951I975313 3 Magic squaresIRow (and column) sum is 15 ICenter entry must be 55 Iwhatever numbers go in a diagonal spot must be in 3 differentpartitions of 15I1 can t be in a diagonal:1+(6+8)= 15,1+(5+9)=15I3 can t be in a diagonal:3+(7+5), 3+(8+4)I951I975313 3 Magic squaresWe have:97531 IWe know 6 and 8 can t go in the top row (9+6=15, 9+8=17)IWe need to put 2 and 4 in top row, but49753?