Puzzle: A Cross of Cards

Puzzles can be fun ways to idle away a stormy afternoon. Or they can be doorways into the world of math, invitations to explore.

And puzzles are great for teaching, too!

Here is a puzzle from a master of mathematical ingenuity…

Dudeney’s Cross of Cards

In this case we use only nine cards — the ace to nine of diamonds. The puzzle is to arrange them in the form of a cross, exactly in the way shown in the illustration, so that the pips in the vertical bar and in the horizontal bar add up alike.

In the example given it will be found that both directions add up 23.

What I want to know is, how many different ways are there of rearranging the cards in order to bring about the result ?

It will be seen that, without affecting the solution, we may exchange the 5 with the 6, the 5 with the 7, the 8 with the 3, and so on. Also we may make the horizontal and the vertical bars change places. But such obvious manipulations as these are not to be regarded as different solutions. They are all mere variations of one fundamental solution.

Now, how many of these fundamentally different solutions are there?

The pips need not, of course, always add up 23.

[From Amusements in Mathematics, H. E. Dudeney, Thomas Nelson and Sons, 1917.]

Play with the Puzzle

When you solve a mathematical puzzle, that’s never the end of the story. You can always find more ways to play with the ideas.

For example, Dudeney’s next puzzle asks a couple of new questions:

• How many different ways can you arrange the cards in a solution without making a fundamentally different answer?
• How many total possibilities are there to solve the puzzle, counting every arrangement as different?

And those two questions led me to wonder:

• How many ways could you arrange the cards in this cross pattern, without worrying about their sums?
• If you put cards into the cross pattern at random, what is the chance your arrangement would solve the puzzle?

What questions can you ask?