If *The Roses of Heliogabalus* (1888) is any guide, Sir Lawrence Alma-Tadema (1836-1912) thought that 222 is a special number. But his painting doesn’t exhaust its secrets. To get to another curiosity of 222, start with 142857. As David Wells puts it in his *Penguin Dictionary of Curious and Interesting Numbers *(1986), 142857 is a “number beloved of all recreational mathematicians”. He then describes some of its properties, including this:

142857 x 1 = 142857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

142857 x 6 = 857142

The multiples are cyclic permutations: the order of the six numbers doesn’t change, only their starting point. Because each row contains the same numbers, it sums to the same total: 1 + 4 + 2 + 8 + 5 + 7 = 27. And because each row begins with a different number, each column contains the same six numbers and also sums to 27, like this:

1 4 2 8 5 7

+ + + + + +

2 8 5 7 1 4

+ + + + + +

4 2 8 5 7 1

+ + + + + +

5 7 1 4 2 8

+ + + + + +

7 1 4 2 8 5

+ + + + + +

8 5 7 1 4 2

= = = = = =

2 2 2 2 2 2

7 7 7 7 7 7

If the diagonals of the square also summed to the same total, the multiples of 142857 would create a full magic square. But the diagonals don’t have the same total: the left-right diagonal sums to 31 and the right-left to 23 (note that 31 + 23 = 54 = 27 x 2).

But where does 142857 come from? It’s actually the first six digits of the reciprocal of 7, i.e. 1/7 = 0·142857… Those six numbers repeat for ever, because 1/7 is a prime reciprocal with maximum period: when you calculate 1/7, all integers below 7 are represented in the remainders. The square of multiples above is simply another way of representing this:

1/7 = 0·142857…

2/7 = 0·285714…

3/7 = 0·428571…

4/7 = 0·571428…

5/7 = 0·714285…

6/7 = 0·857142…

7/7 = 0·999999…

The prime reciprocals 1/17 and 1/19 also have maximum period, so the squares created by their multiples have the same property: each row and each column sums to the same total, 72 and 81, respectively. But the 1/19 square has an additional property: both diagonals sum to 81, so it is fully magic:

01/19 = 0·__0__ 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 __1__…

02/19 = 0·1 __0__ 5 2 6 3 1 5 7 8 9 4 7 3 6 8 __4__ 2…

03/19 = 0·1 5 __7__ 8 9 4 7 3 6 8 4 2 1 0 5 __2__ 6 3…

04/19 = 0·2 1 0 __5__ 2 6 3 1 5 7 8 9 4 7 __3__ 6 8 4…

05/19 = 0·2 6 3 1 __5__ 7 8 9 4 7 3 6 8 __4__ 2 1 0 5…

06/19 = 0·3 1 5 7 8 __9__ 4 7 3 6 8 4 __2__ 1 0 5 2 6…

07/19 = 0·3 6 8 4 2 1 __0__ 5 2 6 3 __1__ 5 7 8 9 4 7…

08/19 = 0·4 2 1 0 5 2 6 __3__ 1 5 __7__ 8 9 4 7 3 6 8…

09/19 = 0·4 7 3 6 8 4 2 1 __0__ __5__ 2 6 3 1 5 7 8 9…

10/19 = 0·5 2 6 3 1 5 7 8 __9__ __4__ 7 3 6 8 4 2 1 0…

11/19 = 0·5 7 8 9 4 7 3 __6__ 8 4 __2__ 1 0 5 2 6 3 1…

12/19 = 0·6 3 1 5 7 8 __9__ 4 7 3 6 __8__ 4 2 1 0 5 2…

13/19 = 0·6 8 4 2 1 __0__ 5 2 6 3 1 5 __7__ 8 9 4 7 3…

14/19 = 0·7 3 6 8 __4__ 2 1 0 5 2 6 3 1 __5__ 7 8 9 4…

15/19 = 0·7 8 9 __4__ 7 3 6 8 4 2 1 0 5 2 __6__ 3 1 5…

16/19 = 0·8 4 __2__ 1 0 5 2 6 3 1 5 7 8 9 4 __7__ 3 6…

17/19 = 0·8 __9__ 4 7 3 6 8 4 2 1 0 5 2 6 3 1 __5__ 7…

18/19 = 0·__9__ 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 __8__…

First line = 0 + 5 + 2 + 6 + 3 + 1 + 5 + 7 + 8 + 9 + 4 + 7 + 3 + 6 + 8 + 4 + 2 + 1 = 81

Left-right diagonal = 0 + 0 + 7 + 5 + 5 + 9 + 0 + 3 + 0 + 4 + 2 + 8 + 7 + 5 + 6 + 7 + 5 + 8 = 81

Right-left diagonal = 9 + 9 + 2 + 4 + 4 + 0 + 9 + 6 + 9 + 5 + 7 + 1 + 2 + 4 + 3 + 2 + 4 + 1 = 81

In base 10, this doesn’t happen again until the 1/383 square, whose magic total is 1719 (= 383-1 x 10-1 / 2). But recreational maths isn’t restricted to base 10 and lots more magic squares are created by lots more primes in lots more bases. The prime 223 in base 3 is one of them. Here the first line is

1/223 = 1/22021_{3} = 0·

0000100210210102121211101202221112202

2110211112001012200122102202002122220

2110110201020210001211000222011010010

2222122012012120101011121020001110020

0112011110221210022100120020220100002

0112112021202012221011222000211212212…

The digits sum to 222, so 222 is the magic total for all rows and columns of the 1/223 square. It is also the total for both diagonals, so the square is fully magic. I doubt that Alma-Tadema knew this, because he lived before computers made calculations like that fast and easy. But he was probably a Freemason and, if so, would have been pleased to learn that 222 had a link with squares.