More Fun With Oxygen Deprivation

So there we were at lunchtime doing Russians when someone said, “Figure out the one number between one thousand and two thousand that isn’t a sum of consecutive integers.”

By “doing Russians” I don’t mean we were pounding vodka drinks nor do I mean we were doggin’ up one of the non-Korean massage parlors. Russians are a weightlifting thing where you stand there and hold (say) forty-five pound barbells and raise them to your shoulders, first one, then the other, back and forth, kind of like you’re working a line of tequila shooters only you’re not. The tale is the Russian military determined it to be the most effective way to build upper arm strength, and I suppose they would know, since when I picture the Russian military I picture a brutal scenario of hard physical labor only occasionally relieved by a nice restful beating. The idea was to do five reps of these, repeated twenty times, the rest period being when other guys were taking their turns.

Quite the merry crew, what.

So to add to the hilarity one of them (who speaks Russian, by the way), said, “You know, most numbers can be expressed as a sum of consecutive integers, right?”

“What?”

“Like fifteen is four plus five plus six, and twenty-one is six plus seven plus eight, and, um.”

“Yeah, and ten is one and two and three and four.”

“Right.”

“So what?”

“So it’s like a mind puzzle. There’s only one number between one thousand and two thousand that can’t be got that way. The trick is to figure out which one.”

“Really? Just one?”

“Yup.”

“How do they know?”

A shrug and a smile. You know how that is.

So we’re pounding these weights up and down and up and down, and some of us have gotten sort of quiet. I don’t know if it’s an engineer thing or what, but there’s something about a puzzle, about something that you know can be figured out if you just discover the right approach. This is especially attractive if your brain is losing ground against physical exertion and your ability to deal with complexities such as proper English or interpersonal relationships or walking and chewing gum at the same time is steadily eroding. Maybe it’s just me but when my mind is starting to darken out, it sometimes turns to mathematics, maybe as a sort of last gasp effort to retain something of itself, some kind of connection to something more than pure animal pain. Prisoners thrown into solitary have been known to go over old math lessons to keep their sanity. I imagine former POW John McCain can relate to this whenever he’s stuck on a plane with Chris Matthews.

“So,” I said, wheezy from my last effort. “All odd numbers are out, cause any odd is the sum of two consecutive numbers.”

“Right.”

“And, um, any even number that can be divided by three is out, cause, like, forty-nine and fifty and fifty-one are one hundred fifty.”

“Okay.”

I figured that idea extended to all like numbers.

“So too if it can be divided by five, or any odd number, cause, you know.” I was thinking of an easy example like twenty five, because …

25 / 5 = 5

… and …

5 + (4 + 6) + (3 + 7) = 25

… or in a similar vein,

56 = 7 * 8 = 8 + 8 + 8 + 8 + 8 + 8 + 8

= 8 + (7 + 9) + (6 + 10) + (5 + 11)

= 5 + 6 + 7 + 8 + 9 + 10 + 11

That much was clear. Obviously there was some formulaic way to discover the answer. By now, after several sets of Russians and a time or two on the tummy-cruncher bench, it was pretty clear my brain was depraved on account it was deprived. I couldn’t stop.

“Ooh! So, huh,” I said. “So since the sum of integers from one to n is n squared plus n over two. And the sum of some arbitrary consecutive series is that minus the lower end series you didn’t use …”

“Uh …” he said.

“… so we got,” I said, “ …

( (n² + n) / 2 ) - ( (m² + m) / 2 )

… to get any good number, in other words the number we want is the one that can’t be found with this equation. So all we gotta do – I need some paper for this – is we take this equation and make m less than n and set it up with some algebra …”

“Dude.”

“… and find the number for which it can’t be made to work …”

“Dude, you’re making this way too complicated.”

“I am? Oh. Yeah. I do that with everything.”

But a couple of the others had been puzzling away on their own and offering thoughts on the solution (“It’s not, like, something stupid like fifteen hundred, is it?”) but gotten nowhere, and I wasn’t going to quit either, not yet. I was actually getting tired from this bizarre workout (we had never done this particular one before) and my lower back was starting to hurt, probably because I wasn’t doing it right, so I bent over and touched my toes to stretch my back out and thought, well, it seems a number is a sum of consecutives so long as if you keep dividing it in half you eventually reach an odd number and can’t divide anymore. I’m not sure what subconscious processing of numbers led to that insight, but it was related to my intuition that the number in question had to be even and be summed with a series that does not have an odd number in the center of it. In other words, if you keep dividing this hypothetical number, you will never reach an odd number, cause if you do, a series can be constructed around that odd number that adds back up to where you started.

Something like that. It isn’t easy to reconstruct thoughts that took place in a state of physical stress and were never meant to be articulated in human speech anyway.

Anyway, from that, I suddenly blurted out the answer, the one integer between one thousand and two thousand that is not the sum of consecutive integers. The guy who posed the question smiled, and everyone else asked how the hell I did that. I didn’t really know, of course. After the exercise some of us went for a run and got REALLY tired, and it’s probably a small mercy that I can’t remember thinking about anything. I probably didn’t. I think my brain went into one of those impenetrable early afternoon comas because at one o’clock I had a meeting to go to anyway.