## Doubling Back

Two and two are four,

Four and four are eight,

Eight and eight are sixteen,

Sixteen and sixteen are thirty two.

Two and two are four…

I heard that counterpoint background to Danny Kaye’s portrayal of Hans Christian Andersen only once, at about the age of 10, when our neighbors invited us over to watch the movie on TV. For a little song like that to a little math-bent mind like mine always was, once was enough to have me singing the tune throughout my childhood. Sometimes to this day I can be heard singing it when the mood hits me.

When I felt like irritating my brothers or others, I would not continue it as in the movie, restarting each cycle at two each time. Instead, my second verse would go, “32 and 32 are 64, 64 and 64 are 128, 128 and 128 are 256, 256 and 256 are 512.” And then to be seriously demented and obnoxious, I was known to have gone higher from there, “512 and 512 are 1,024, …” and so on. I once took that out over ten verses on a band bus before I got shrieked at to have mercy on the other band members.

In 12th grade, I sat for a special test co-sponsored by the Society of Actuaries, little knowing then where my future career would take me. One question on that multiple-choice test was to choose the last digit of 2 to the 400th power minus 1. Raise your hands if you tried to multiply 2 by itself 400 times, go ahead, don’t be shy. Now, how many of you tried a little shortcut, via 2 times 2 is 4, 4 times 4 is 16, 16 times 16 is 256, 256 times 256 is . . . uh . . .?

I selected the correct answer almost immediately, and felt it no genius in doing so. We want only the final digit, so all other digits can be dropped out of our multiplications. So we proceed – 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6 . . . and even a near-dunce head such as mine doesn’t take more than that many cycles to figure out the pattern. So any power of 2 that is divisible by 4 will have 6 as its last digit, and 6 minus 1 is 5, which was the answer to the test’s question.

Near-dunce me, it took me over 40 years, until just today, for me to realize that the pattern that gave me the answer to that problem in 12th grade was exactly the same pattern I’d been singing since seeing the Hans Christian Andersen movie. Instead of continuing my song verses with the higher numbers, all I needed to do was to sing the song with only the final digits —

Two and two are four,

Four and four are eight,

Eight and eight take us back to six,

Six and six come back round to two.

Two and two are four…

Virtually every single thought I have ever had concerning mathematics has felt similar to me as this pattern. Right down to how slow my own head is for working it out, other than for the vision of it inside my head.

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