Prime numbers tie into all sorts of different Magic Squares. Here’s a peek at how the relationship between the two plays out, based on the size and type of Magic Square.

An Almost Prime Magic Square

The great English puzzlist Henry Ernest Dudeney came up with the following 3 x 3 magic square consisting almost entirely of primes way back in 1917. The Magic Sum is 111. However due to the fact that the number 1 is not prime, it is not strictly a prime magic square.

art2img1

Mathematical Recreations and Essays (Rouse Ball 1939)

A Prime Magic Square

This has a magic sum of 177. This magic square has the left diagonal triples descending by a constant value of 42, along with the right diagonal triples ascending by a constant value of 12.

art2img2

Amusements in Mathematics (H. E. Dudeney 1958)

Arithmetic Prime Magic Square

This magic square has a magic sum of 3117.

art2img3

What makes this particular prime magic square so interesting is when we arrange their values in numerical order.

199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879

What do you notice? (click to see below)

The difference between them is the same: 210.

Amusements in Mathematics (H. E. Dudeney 1958)

Ortho Magic Square

An orthomagic square has been described as one in which only the rows and columns sum to the same amount, not the diagonals. The Ortho Magic Square below is not made up of primes, however, each of numbers in it are expressed as the square of primes.

art2img4

4 x 4 Prime Magic Square

The following prime magic square has a magic sum of 258.

art2img5

This magic square has a peculiar property; can you identify what it is?

Hint: write out the primes in numerical order.

Click to reveal the answer

The primes are consecutive.

Journal of Recreational Mathematics (Vol 14:2, 1981-2)

5 x 5 Prime Magic Square

The following prime magic square has a magic sum of 313. It has the same property as the previous 4 x 4 prime magic square.

art2img6

Journal of Recreational Mathematics (Vol 14:2, 1981-2)

6 x 6 Prime Magic Square

The following prime magic square has a magic sum of 484. It has the same property as the previous two prime magic squares.

art2img7

Journal of Recreational Mathematics (Vol 14:2, 1981-2)

7 x 7 Prime Magic Square

The following prime magic square has a magic sum of 26, 627!

art2img8

If this isn’t already a remarkable feat, there is another aspect of this extraordinary prime magic square.

If we remove the “units” off each of the primes, (i.e. 71 becomes 7 etc.), we generate a 7 x 7magic square with a magic sum of 2,760.

art2img9

Madachy’s Mathematical Recreations (Joseph Madachy 1979)