More Than One Way to Solve It

Photo by Eirik Newth via flickr.

In a lazy, I-don’t-want-to-do-school mood, Princess Kitten was ready to stop after three math problems. We had gotten two of them correct, but the last one was counting the ways to paint a cube in black and white, and we forgot to count the solid-color options.

For my perfectionist daughter, one mistake was excuse enough to quit. She leaned her head against me as we sat together on the couch and said, “We’re done. Done, done, done.” If she could, she would have started purring — one of the most manipulative noises known to humankind. I’m a soft touch. Who can work on math when there’s a kitten to cuddle?

Still, I managed to squeeze in one more puzzle. I picked up my whiteboard marker and started writing:

DONE
DOEN
DNOE
DENO
DNEO
ONED
ODNE

Let Me Count the Ways

We’re working our way through the counting lessons in Competition Math. We’ve done Venn diagrams, triangular numbers, and the Fundamental Counting Principle, and today we studied casework (counting the various possibilities for each possible case). But I figured this problem was review, a fairly basic application of the FCP:

4 choices for the first letter.
Times 3 choices for the second letter.
Times only 2 choices left for the third…

Naturally, that’s not the way Kitten figured it out. Why does it still surprise me when she thinks differently than I do? I should expect it by now.

Try Switching the Letters

Kitten pointed at my list.

“If we have 5 ways to start with D, then we could switch the D and the O. So then we’d have all the same ways to write it, except with the D replacing the O and the O replacing the D.

“Or we could switch the D and the E, or the D and the N. That means we really have 4 times as many ways. It’s simple!”

That made sense. It should work. But we had a problem…

The Answers Have to Match

Using the Fundamental Counting Principle, I calculated 24 ways to arrange the letters. Kitten found 20 solutions with her Letter Switching Principle. But no matter which way we count them, the number of arrangements shouldn’t change.

At least one of us had made a mistake.

I looked back over my list. Sure enough, I had missed one of the ways to start with D:

DONE
DOEN
DNOE
DENO
DNEO
and one more: DEON

Now Kitten’s Letter Switching Principle gave us 6 4 = 24 arrangements.

Success!