Chimbumu and Chikwama get out of jail

by Dinoj Surendran

It was the year 2162 and the height of the Cold War. The twin powers of Siberia and Antarctica were pitted in a titanic struggle over who controlled an insignificant bit of land in the middle of the Pacific. But we aren't interested in that. Our attention is focused on Cell No. 224 in the muggy quarters of the East Siberian Prison near the picturesque city of Vladisputnik. Its residents are the legendary Chimbumu and Chikwama.

Chimbumu was clearly excited.

"Big news! They're going to release some of us!"

"Release?", Chikwama racked his cerebral archives for a word he hadn't thought of in years. "You're kidding! Who's getting released?"

"I don't know. But I know how they'll do it. Let's say the number of cells here is n. Tomorrow, the jailer will make n passes along the cells like this:

• In the first pass, he'll open all locks.
• In the second pass, he'll lock every second lock, ie those to cells 2, 4, 6,...
• In the third pass, he'll unlock every third lock, and so on.
• So in the k-th pass, he'll turn every k-th lock, ie k, 2k, 3k,...

Question is, whose locks will be open at the end of all that?''

"My question is, how will one jailer do all that?"

Oh, I'm sure they'll use all the jailers. But the instructions will be followed explicitly. I suspect it's an attempt to teach them basic counting skills!"

"Hmmmm. Let's get serious. Let us consider the c-th cell. It is locked to begin with. It is unlocked on the first pass, and if c is divisible by 2, it is locked on the second pass. Again, if it is divisible by 3, it will be un/locked on the third pass..."

"Ah! So if c is divisible by k, its door will be un/locked on the k-th pass!"

"Right. Now we want the c-th cell to be open at the end so how many times must its lock be turned?"

"An odd number of times! And the lock is turned on the k-th pass only when c is divisible by k --- so c must have an odd number of divisors!"

"Exactly. And which numbers have an odd number of divisors?"

"Beats me."

And thus it was, since neither knew how to proceed from there on, that C & C went to their bunks with heavy hearts, cursing that their ignorance should be their final obstacle to freedom. Thus it was also that Chimbumu, failing to sleep, turned his mind to the problem. His mind produced the following table:

Surprise no.1: very few numbers had an odd number of divisors

Surprise no 2: these numbers were 1, 4, 9, 16 --- all square numbers!

Chimbumu fell out of bed in his rush to wake up his slumbering cellmate. He eventually succeeded, and they were soon discussing this latest development.

"Of course it must be so! Because for non-square numbers, the divisors always pair up. For example 12 has divisors 1 and 12, 2 and 6, 3 and 4. This isn't the case for a square number --- its root can't pair with anything."

Things were of course easy from then on. The two jailbirds crossed in to the empty neighbouring cell, 225 = 15^2 --- they had fortunately tunneled into it a few years previously, and the next morning, they saw the light of the sun again for the first time in ten years.

Sadly, it was also the last time in ten years, for Chimbumu had misheard the message --- those who finally left their cells were not supposed to be released, but executed...

The author stole the idea for this article from that Classic of the Dolciani Mathematics Expositions, "Mathematical Gems I" by Ross Honsberger.