## Posts tagged ‘Armstrong numbers’

### Therapeutic Numbers

I was lying on my left side, my right leg awkwardly bent so that my right foot was flat on the floor in front of me, and my left leg was extended straight out underneath my bent right leg. There was a weight strapped around my left ankle, and I was lifting my left leg as high as I could. “How many?” I asked.

“Thirty,” said my physical therapist.

I’m not sure if she heard me gulp. I had only done eight so far, and already my thigh was screaming.

But that was nothing compared to the guy next to me. He was lying face-down on a table, his head and arms hanging off of one end. In each hand was a dumbbell, and he had to rotate his shoulder joint until his arms were parallel to the ground. After his first few, he asked, “How many?” She told him 30, too.

He did a few more, and his grunts were getting louder. “How many?” he asked again, but now with an air of incredulity.

“One-hundred fifty-two,” our therapist said. “That’s *always* the answer the *second* time you ask.” She smiled, then she looked at me. “I’m not sure why 152 is the number I pick.” It seemed reasonable that she’d want to explain her choice to me. After all, I am a numbers nerd. (Not a dweeb, geek, or dork. See below.) But then I realized she doesn’t even know what I do.

“It’s a fine number to pick,” I said. “After all, it’s evenly divisible by the sum of its digits: 152 / (1 + 5 + 2) = 19.”

She squinted a bit, and she raised one eyebrow slightly. I was undeterred.

“But 153 might be a better choice,” I continued. “It’s the sum of the first 17 counting numbers: 1 + 2 + 3 + … + 17 = 153. And if you raise each of its digits to the third power and then add them, you get 153 again: 1^{3} + 5^{3} + 3^{3} = 153.”

She now raised both eyebrows. Her head shook a little as she asked, “You just *know* that?”

“Yes,” I said. “And I often wonder how much useful information I could keep in my head if it weren’t filled with all this trivia about numbers.”

Then there was a long, silent pause. It probably would have been uncomfortable to a less mathematical, more socially adept individual. But not me. However, I felt bad when my therapist started to squirm, so I continued.

“What’s exceptionally cool, though, is that if you take any three-digit multiple of 3, and then add the third power of its digits, and then add the third power of the digits of the result, and keep doing that, you’ll always get back to 153.”

There was another long, silent pause.

The shoulder guy next to me finished his exercises. “What next?” he asked.

“How about some dumbbell presses,” she suggested.

“How many?”

She looked at me. “15*3*,” she said, with a little extra emphasis on the three.