### Ultra magic 8x8 square

Ultra magic versus most perfect

The ultra magic 8x8 square is just like the most perfect magic square panmagic, 2x2 compact and 1/2 rows/columns give 1/2 of the magic sum; 1/2 diagonals give not 1/2 of the magic sum, but the ultra magic 8x8 square is symmetric instead. The challenge was to use a 4x4 magic square to construct an ultra magic 8x8 square. See below my 3 attemps.

Ultra magic 8x8 square (1)

Zie See below the pattern of a semi panmagic 4x4 square (= group 3 of the 880 magic 4x4 squares excluding rotating and/or mirroring). This square has the same structure as the famous Dürer magic square.

The
sum of each colour is the lowest plus the highest number, that is (for the 8x8 magic square) 1 + 64 = 65.

Use 4 binary grids to construct the 4x4 Dürer magic square:

 1x digit 2x digit 4x digit 8x digit +1 Magic 4x4 square 0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 15 8 10 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 14 4 11 5 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 12 6 13 3 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 7 9 2 16

Putting the binary grids in random order give 4x3x2x1 is 24 possibilities. Choosing in each grid 0 or 1 give 2x2x2x2 is 16 possibilities. In total there are including rotating and/or mirroring 24 x 16 is 384 semi panmagic 4x4 squares of group 3.

 1x number from 2x 4x4 & 2x inverse 1 15 8 10 1 15 8 10 14 4 11 5 14 4 11 5 12 6 13 3 12 6 13 3 7 9 2 16 7 9 2 16 16 2 9 7 16 2 9 7 3 13 6 12 3 13 6 12 5 11 4 14 5 11 4 14 10 8 15 1 10 8 15 1 + 16x number from Sudoku grid 0 3 1 2 1 2 0 3 3 0 2 1 2 1 3 0 2 1 3 0 3 0 2 1 1 2 0 3 0 3 1 2 0 3 1 2 1 2 0 3 3 0 2 1 2 1 3 0 2 1 3 0 3 0 2 1 1 2 0 3 0 3 1 2 = ultra (pan)magic 8x8 square 1 63 24 42 17 47 8 58 62 4 43 21 46 20 59 5 44 22 61 3 60 6 45 19 23 41 2 64 7 57 18 48 16 50 25 39 32 34 9 55 51 13 38 28 35 29 54 12 37 27 52 14 53 11 36 30 26 40 15 49 10 56 31 33

This 8x8 magic square is panmagic, symmetric in each 4x4 sub-square and 1/2 row/ column/diagonal gives 1/2 of the magic sum. If you divide the 8x8 magic squares into 2x2 sub-squares, the digits of the 2x2 sub-squares give 1/2 of the magic sum. The 8x8 magic square is not 2x2 compact, because not each random chosen 2x2 sub-square gives 1/2 of the magic sum. The magic 8x8 square is symmetric in each 4x4 sub-square.

Ultra magic 8x8 square (2)

Construct a 4x4 semi panmagic square of group 2:

Construct the 4x4 semi panmagic square of group 2 by using binary grids.

 1x digit 2x digit 4x digit 8x digit +1 Mag. 4x4 square 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 11 8 14 1 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 6 16 3 9 1 1 0 0 1 0 0 1 0 0 1 1 1 0 1 0 12 2 13 7 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 15 5 10 4

P
uzzle and find the right Sudoku grid:

 1x number from 2x 4x4 & 2x inverse 1 11 8 14 1 11 8 14 6 16 3 9 6 16 3 9 12 2 13 7 12 2 13 7 15 5 10 4 15 5 10 4 4 10 5 15 4 10 5 15 7 13 2 12 7 13 2 12 9 3 16 6 9 3 16 6 14 8 11 1 14 8 11 1 + 16x number from Sudoku grid 0 2 1 3 1 3 0 2 1 3 0 2 0 2 1 3 2 0 3 1 3 1 2 0 3 1 2 0 2 0 3 1 2 0 3 1 3 1 2 0 3 1 2 0 2 0 3 1 0 2 1 3 1 3 0 2 1 3 0 2 0 2 1 3 = ultra (pan)magic 8x8 square 1 43 24 62 17 59 8 46 22 64 3 41 6 48 19 57 44 2 61 23 60 18 45 7 63 21 42 4 47 5 58 20 36 10 53 31 52 26 37 15 55 29 34 12 39 13 50 28 9 35 32 54 25 51 16 38 30 56 11 33 14 40 27 49

This 8x8 magic square is panmagic, symmetric in each 2x2 sub-square and 1/2 row/ column/diagonal gives 1/2 of the magic sum. If you divide the 8x8 magic squares into 2x2 sub-squares, the numbers of the 2x2 sub-squares give 1/2 of the magic sum. The 8x8 magic square is not 2x2 compact, because not each random chosen 2x2 sub-square gives 1/2 of the magic sum. The magic 8x8 square is symmetric in each 2x2 sub-square.

Groups 4, 5 and 6 of the magic 4x4 square are also semi panmagic. Try it yourself to construct an ultra (pan)magic 8x8 square by using a 4x4 semi panmagic square of group 4, 5 and/or 6.

Ultra magic 8x8 square (3)

The previous 8x8 magic squares are not completely ultra magic, because these magic squares are not 2x2 compact. It is now time to find a valid ultra magic 8x8 square. Only suitable 4x4 magic square to construct a 2x2 compact 8x8 magic square is the panmagic square (group 1).

I have put in the left top quarter of the first 8x8 grid a 4x4 panmagic square. In the other quarters of the 8x8 grid I have put the shifted versions of the 4x4 panmagic square. Notify that the grid is symmetric.  I have puzzled a second Sudoku grid, which is symmetric as well and leads to a magic 8x8 square with all the numbers from 1 up to 64.

1x number from [shifted version] panm.4x4

 1 15 6 12 2 16 5 11 14 4 9 7 13 3 10 8 11 5 16 2 12 6 15 1 8 10 3 13 7 9 4 14 3 13 8 10 4 14 7 9 16 2 11 5 15 1 12 6 9 7 14 4 10 8 13 3 6 12 1 15 5 11 2 16

+ 16x number from Sudoku grid

 0 3 0 3 1 2 1 2 3 0 3 0 2 1 2 1 0 3 0 3 1 2 1 2 3 0 3 0 2 1 2 1 2 1 2 1 3 0 3 0 1 2 1 2 0 3 0 3 2 1 2 1 3 0 3 0 1 2 1 2 0 3 0 3

= ultra (pan)magic 8x8 square

 1 63 6 60 18 48 21 43 62 4 57 7 45 19 42 24 11 53 16 50 28 38 31 33 56 10 51 13 39 25 36 30 35 29 40 26 52 14 55 9 32 34 27 37 15 49 12 54 41 23 46 20 58 8 61 3 22 44 17 47 5 59 2 64

This 8x8 magic square is panmagic, symmetric and 2x2 compact and each 1/2 row/ column gives 1/2 of the magic sum. The most easy method to construct an ultra magic 8x8 square is the basic key method (symmetric).

N.B.:  Use the basic key method and try the key below (and you get the same result as ultramagic 8x8 square (2), but this time fully 2x2 compact and 1/2 rows/columns/ diagonals don't give 1/2 of the magic sum):

 1 1 4 4 6 6 7 7 8 8 5 5 3 3 2 2

 1 57 4 60 6 62 7 63 8 64 5 61 3 59 2 58 25 33 28 36 30 38 31 39 32 40 29 37 27 35 26 34 41 17 44 20 46 22 47 23 48 24 45 21 43 19 42 18 49 9 52 12 54 14 55 15 56 16 53 13 51 11 50 10

Ultra magic 8x8 square (4)

And if you like a dessert. It is also posible to construct an 8x8 magic square which is panmagic, not symmetric, 3x3 (the numbers in the 4 corners) compact instead of 2x2 compact and 1/2 rows/columns/diagonals give 1/2 of the magic sum.

Just like ultra magic (3) take a 4x4 panmagic square, but change it to get a 3x3 (the numbers in the 4 corners) compact instead of 2x2 compact result by swapping rows and columns systematically.

 1 15 6 12 1 6 15 12 1 6 15 12 14 4 9 7 14 9 4 7 11 16 5 2 11 5 16 2 11 16 5 2 14 9 4 7 8 10 3 13 8 3 10 13 8 3 10 13

I have put in the left top quarter of the first 8x8 grid a 4x4 square which is 3x3 compact. In the other quarters of the 8x8 grid I have put the shifted versions of the 4x4 square. I have puzzled a second Sudoku grid, which is 3x3 compact as well and leads to a magic 8x8 square with all the numbers from 1 up to 64.

1x number from [shifted version of] 3x3 compact 4x4

 1 6 15 12 5 2 11 16 11 16 5 2 15 12 1 6 14 9 4 7 10 13 8 3 8 3 10 13 4 7 14 9 11 16 5 2 15 12 1 6 1 6 15 12 5 2 11 16 8 3 10 13 4 7 14 9 14 9 4 7 10 13 8 3

+ 16x number from Sudoku grid

 0 3 0 3 1 2 1 2 3 0 3 0 2 1 2 1 3 0 3 0 2 1 2 1 0 3 0 3 1 2 1 2 0 3 0 3 1 2 1 2 3 0 3 0 2 1 2 1 3 0 3 0 2 1 2 1 0 3 0 3 1 2 1 2

= panmagic & 3x3 compact 8x8 square

 1 54 15 60 21 34 27 48 59 16 53 2 47 28 33 22 62 9 52 7 42 29 40 19 8 51 10 61 20 39 30 41 11 64 5 50 31 44 17 38 49 6 63 12 37 18 43 32 56 3 58 13 36 23 46 25 14 57 4 55 26 45 24 35

See the download below with all 4 'ultra magic' 8x8 squares and you can check if all formulas give a valid result and all numbers from 1 up to 64 are in the magic square.

8x8, Ultra magic square.xls