As the year draws to a close, plenty of Grey Matters favorites seem to be putting in one last 2007 appearance. For example, I've recently found a new online video featuring Arthur Benjamin.

Over at Online Visions, Werner Miller is back with a mathematical twist on an old idea, which he called A Holey Number. Werner Miller seems to have a particular fondness for creative approaches to magic squares, as his Square Bet and Age Cube demonstrate.

The first simple challenge of A Holey Number is to put four puzzle pieces together to make a magic square. This is easily done, and results in a magic square whose rows, columns and diagonals add up to 30. The four pieces are then picked up, mixed, and turned over. The same challenge is given. This time, there's a big surprise. The magic square still totals 30, but when the puzzle is assembled, the 0 is replaced by an empty space!

As mentioned in the first paragraph of the article, this is presented as a puzzle or paradox. If you want to, as Werner Miller puts it, “have a strange piece of story-telling magic”, then you can alter the presentation. For a puzzle-based routine like this, a great place to start would be the works of Robert Neale (especially The Magic Mirror) or Stewart James. My Memorable Magic series of posts could also be helpful here.

As it happens, the pattern used in A Holey Number can be arranged on a standard 15 puzzle. Due to this, you might consider mixing A Holey Number with my 40 30s 4 15 routine (video clip).

One way to present this would be as a farcical race, to see who could be the first to see who can make a magic square with 15 numbers. Since you as the performer has to work with 15 sliding tiles (and get them into the 11, empty space, 7, 12 pattern), and the spectator has to work with only the four pieces of the red side. After they've obviously already finished, you finish arranging the 15 puzzle and facetiously declare victory. When it's pointed out that the spectator finished first, you hesitate, while trying to think of a way out of this. Then, you remember that you specified that the magic square should be made with 15 numbers, showing that you really did use only 15, and their four-piece puzzle involved 16 numbers (uh, yeah, that's it! That's the ticket!). Give them a second chance to beat you, pick up the four pieces, turn them over to the blue side, and have them arrange those pieces. Since they have shown how quickly they can do it, and it was still faster than your 15 puzzle time, you finally admit they're the winner, leaving them puzzled as to why the 0 is an empty space now.

If you have any other presentation ideas for A Holey Number, I'd love to hear about them in the comments!