How to Construct a Forcing Matrix

History

The forcing matrix concept was first given magical application by Walter Gibson in 1938 in a strictly informational description. The actual forcing modification was put into print by Maurice Kraitchik in 1942. Other notables who subsequently worked with it include Mel Stover, Stewart James, Martin Gardner, Howard Lyons, Leslie May, Sam Dalal, Paul Hallas, Max Maven, and Richard Busch.

The forcing matrix does not have to be a grid of numbers (check out the elegant Quintasense in T.A. Waters’ Mind, Myth and Magick, p. 279), but that is perhaps the most “open” way of doing it.

An Example

Here’s a typical forcing matrix … I’ve chosen a 5x5 one, as it seems to be a good size for this, but the methodology will work for any square matrix (same number of rows and columns):
13 11 14 19
10 15
10 11 16
17 12 15 18 23
14

To try it out, circle any number, and then cross out the remaining numbers in the same row (horizontally) and column (vertically). Then circle another number (one not already eliminated), and again strike out the numbers above, below, to the left, and to the right of same. Repeat until all numbers are either circled (there will eventually be five) or crossed out. Add the chosen (circled) numbers together. Now concentrate … I sense that the total will be … wait a second … fifty-seven!

There are more deceptive approaches than the above method of choosing the numbers. Max Maven suggested the use of coloured pencils (for a 5x5 square, five colours are needed; draw a differently-coloured line through each column; repeat for the rows; add the numbers where like colours intersect). I have often used the following presentation: pick an interesting word with the same number of (different) letters as the rank of the matrix (say “MAGIC” for the above); write this word across the top, a letter over each column; have the participant rearrange the letters in any order desired, and write them down the left side, a letter beside each row; circle the numbers at the intersection points of matching letters. These alternative presentations avoid the appearance of a diminishing (and thus limited) choice of numbers, suggesting a force. Which, in fact, it is.

It’s best if you actually take the trouble to try this for yourself, before reading on to learn how it works. The result is quite elegant and surprising, even to those with some mathematical sophistication.

Construction

The above example is fine if you want to force the number 57, but what if your plans call for a different result? Here’s how to construct your own matrix, to force a number of your choosing. Again, I’m using a 5x5 matrix to illustrate, but any square configuration will work.

Select any ten numbers (which we will call the seeds) that sum to your desired total (I’ll use 57 again, so you can see how the above matrix was constructed). For example, 6 + 1 + 4 + 7 + 12 + 7 + 3 + 4 + 11 + 2. Write half of those seeds above the columns of the matrix, and the remaining five to the left of the rows. Then fill in the matrix by placing, in each cell, the sum of the two related seeds (the one directly above and the one on the left), as follows:

12
13 11 14 19
10 15
10 11 16
11  17 12 15 18 23
14

Finally, erase the seeds, and you have your forcing matrix. The seed numbers can be of any type (positive, negative, integers, fractions, rational, irrational), and include duplicates, as long as there is one for each column and row, and their sum is the desired force value.

With a little thought, you can now see how a forcing matrix works. Each number in the matrix represents the sum of two seeds. When a number is selected (e.g., circled), its associated pair of seeds is effectively eliminated, because all of the other numbers that were formed by using them are removed from consideration. So the five eventually chosen numbers are the sums of five different pairs of seeds, which is the same as the sum of all ten seeds (i.e., the force number).


If you don’t object to having a zero as one of the grid numbers, it’s possible to use the seeds as entries in the matrix. This enables one to construct the matrix directly, without having to go back and erase the seeds (it also means that two fewer seeds are required). To construct such a matrix, place a zero in any cell, then fill in the rest of the cells in that row and column with the seeds. Finally, fill each remaining cell with the sum of the two seeds whose respective row and column that cell shares. Here are two examples of forcing matrices constructed in such a fashion:

22 10 27 13 7
35 19 15 23 40 12 17 22 10 16
20 25 10 15 20 14
41 25 21 29 46 12 6
25 13 30 11 16 10
force value = 100 force value = 50

In the finished matrix, of course, the seed values would not be highlighted as shown here. This alternative method doesn’t allow quite as much freedom in the selection of seeds (sometimes handy to make the resulting numbers more uniform), but it can be useful when there is a need to quickly generate forcing matrices.


Those interested are welcome to make use of a simple spreadsheet that I have constructed to illustrate the technique, and perform the additions.

If you want to read more about this topic, an excellent reference is Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games, by Martin Gardner (who else?), though what I’ve described here is enough to let you generate all the forcing matrices you’ll ever need. Enjoy!

… Doug Dyment

www.deceptionary.com