A Holey Number
by Werner Miller

Werner Miller is a retired teacher of mathematics, whose hobbies are recreational mathematics and magic. A magical inventor and writer, Werner has produced a prolific volume of work, mostly what he calls “semi-automatic” card tricks based on mathematical principles. He is the author of "Fast von selbst", "Alles Miller oder was", "Ratatouille", more than 300 trick contributions to various magazines and web sites and more than 30 related computer programs. In addition, Werner is also a staff member of the German magazine "Magische Welt", a columnist for the British magazine "The Magician", and regular contributor here at Visions. Werner's first English-language book, Ear-Marked, is available here.

This is merely a puzzle or a paradox, not a trick. If you like it bizarre, feel free to add a suitable patter to have a strange piece of story-telling magic.

Here are the bare bones:

You introduce four irregularly shaped number-printed pieces (Fig. 1).

The spectator is requested to put them together so that the numbers are forming a magic square. The solution is easily found (Fig. 2). The four numbers of each row, each column, each diagonal add up to 30.

The four pieces are turned over (Fig. 3). Again, the problem is to produce a magic square, the same as before in fact.

The solution brings a surprise: It seems that the cell formerly containing the number 0 is completely gone leaving a square hole (Fig. 4)! Nevertheless the sum of each row, each column, each diagonal is still 30.

Do you want to try this oddity yourself? I prepared a template (Fig. 5). You can download it here. Print it out, cut out the four pieces, fold each piece in half, and paste front and back together.

Notes
The basic paradox (to cut a square into four pieces which can be rearranged to form another square with a hole in the center) is very old. See Martin Gardner's "Mathematics, Magic and Mystery", New York 1956, p. 151.

The idea to apply it to a magic square is also not new. Serhiy Grabarchuk's version entitled "Where Is the 5?" using the standard 3x3 magic square is to be found in the web at puzzles.com.

The 4x4 magic square I use is a so-called pandiagonal magic square, i.e. the "broken" diagonals (0 - 9 - 15 - 6, 7 - 2 - 8 - 13, 12 - 5 - 3 - 10, 13 - 4 - 2 - 11, 10 - 15 - 5 - 0, 1 - 8 - 14 - 7) add up to the square's constant, too. This makes it possible to let any number of the square "vanish". For obvious reason I decided on the holey number zero.

Once, Stewart James was asked for advice on how best to create magic. He gave several suggestions, among them this one: to study puzzles. "The simplest puzzle is far cleverer than the most brilliant trick. You can't cheat to arrive at the solution of a puzzle. A magician invariably cheats in order to complete a trick." Think about this advice!

Werner Miller