In the below image we have a magic square of a size 3x3. The magic number for all its rows, columns and both diagonals is 165.
Rotate the grid 180 degrees and all sums still have the magic number 165. Hold a mirror to the edge of the grid and the reflection still adds to 165

Is it possible that this is the only square where this works?
digital magic square

Question 1
Using digits as displayed on a digital clock, how many other magic squares of size 3x3 can be made that hold the rotation or the reflection properties, or both?

Question 2
How many different magic constants can be created using this method?

0,1,2,5,8 can all be used and retain their values
3,4,7 can't be rotaed or mirrored so can't be used.
6,9 can be rotated to create each other but can't be mirrored.


1 Answer 1


Yes without diagonals, No with them.

Since you can only use 0,1,2,5,8 you can try adding two more possibiliteies

Replacing all eights by ones or zeros but then the diagonals wont work

For the diagonals to work you would need

To find 3 digits that add to the middle one times 3 (2+5+8 = 5+5+5) The only other configuration is using [0,1,2] but in a mirror you will have [0,1,5] thus not working for diagonals in a mirror.

to balance the 2 in the mirror, we need a 5 and the only group of 3 digits are [2,5,0], [2,5,1], [2,5,8] , one of them in the question, the other two by replacing 8s with 0s or 1s.

You could create more by allowing more digits do every cell, but then the answer is trivial, just append a 1 to every cell.


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