Nasik cube of 11
These are some pictures I took in 2019 of Andrew Hollingworth Frost’s “Nasik cube of 11” (that is, 11 × 11 × 11) held in the Royal Holloway archives. It had been moved into the newly-built Emily library a year or so before.
A Nasik cube is an m × m × m lattice of numbers in which every row, column, and long diagonal sums to the same number (there’s a little more to it; definitions vary). Wolfram Mathsworld has more detail on magic cubes (and a note about Nasiks).
If, like me, you are not a mathematician, it’s simply a strange and marvellous artefact. The cube, in a wooden frame, is made of sheets of glass with numbers printed on paper glued into position. The unexpected notation — the t you can see in some of the photos — is explained in the accompanying statement.
Frost made this particular cube around 1878. It is accompanied by his notes investigating, amongst other things, plane-covering knight’s tours and a rather splendid 4 × 4 Graeco-Latin square. (This caught my attention because there’s a particular Graeco-Latin square embedded in a puzzle within Planetarium).
Frost was an enthusiastic missionary, and Nasik is the name of the town in India where he was living when he first investigated magic cubes.
A MAGIC CUBE OF ELEVEN.
Constructed by the Rev A.H.Frost, M.A., of St Johns College, Cambridge
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This Cube is an extension of the Magic Squares of the Middle Ages, which consisted of numbers arranged in a certain manner in a square, and were used as amulets, or charms for success in love, war, &c.
Viewed from the face which has 1127 in the right upper corner, this Model contains the consecutive numbers from 1 to 1831 (the cube of 11); the middle number of the series is 666, and these three numbers may be seen on a diagonal of the square on the middle or 6th glass. 33 different square sections of the Cube can be taken, viz., 11 on the glasses, 11 from front to back vertically, and 11 horizontally. Each of these squares contains 121 of these 1381 numbers, and in any square the sum of the 11 numbers in any row, column, or diagonal is the same, viz., 7326, or 11 times 666. This number will also be obtained by adding the 11 numbers in any of the 10 pairs of lines parallel to a diagonal and on opposite sides of it, which together contain 11 numbers, and the 11 numbers on each of the four diagonals of the Cube. The eleven pairs of numbers lying on lines through the central number 666, equidistant from it and on opposite sides of it, have for their sum twice 666, or 1332.
These various summations may be tested at a glance, without actual addition, by affixing at the back of each number its equivalent expressed in relation to elevens instead of tens: e.g., at the back of 1127 we find 935; but here the 3 is 3 elevens, instead of 3 tens, and the 9 is 9 times 121 (11 elevens) instead of 9 hundreds. If the Cube is turned round, the 1331 numbers will be seen in the above altered but equivalent form. In the square now in front, taking, e.g., the column headed 394, and running the eye up its eleven numbers, in the place of units will be seen the eleven numbers 1, 2,..., t, e (t and e denoting ten and eleven), and in the place of the other two digits the eleven numbers 0, 1, 2,..., 9, t. This is the case in all the various rows, columns, &c., their order only varying. The sum therefore is the same in each.
The cube and notebooks were donated to the college archives by Frost’s daughter, Catherine Frost. It is protected by a neat perspex case and is not currently on public view.
Thanks to Royal Holloway’s archives team at the time, specifically Annabel and Ethan, who creatively indulged me with my requests to be shown “interesting things”. At one point I’d clarified that my application to inspect their first edition of Alice in Wonderland was in no way an academic request, but that I simply wanted to geek out over the book’s Tenniel illustrations. I received the most welcoming of reassurances by way of a reply: “Our Reading Room is a safe space for geeking out over books”.