Well, you're in a real pickle. You see, a few years ago, the king decided your life would be forfeit unless you tripled the gold coins in his treasury. Fortunately for you, a strange little man appeared and magically performed the feat. He placed handfuls of coins in and out of a magical bag, and sang a strange rhyme:
“The more gold goes in and more comes out, as sure as I am me. And in again, and out again, and now it’s it times three!”
Incredibly, that tripled the coins and saved your life. Were you grateful? Yes. Were you desperate? Yes. Did you promise him your first-born child in exchange for his help? Yes.
Fast forward to today. No sooner have you given birth to a beautiful baby boy than the little man shows up to claim his prize. You cry and beg him not to take the baby. Softening, he begins, “If you can guess my name—” “Banach-Tarski?” you say. “It’s on the front of your shirt.”
“What! That won’t do. Aha. My bag,” he explains, “increases the number of gold coins placed inside it in a very special way. If I take any number of coins and place them in, more will come out. And if I place those in the bag again, the total that comes out will be three times whatever I began with.”
He takes 13 coins and places them in the bag, then removes the contents. “I’ve used the magic once, not twice,” he says. “Tell me how many coins are in my hand and I’ll have mercy.”
How many coins is he holding?
Pause here to figure it out yourself. Answer in 3
Answer in 2
Answer in 1
The bag’s magic works just like what in mathematics is called a “function,” and it’s convenient in both cases to use an arrow to denote the transformation. We can write what we know like this.
We want to know what goes in this particular blank.
Maybe the bag just multiplies the number of coins by some number. In that case, multiplying by that number twice would be the same as multiplying by 3, which means the multiplier would be the square root of 3. That’s not a whole number, though. And we don’t have bits of gold coins coming out of the bag. Something else is going on.
Well, if filling in the blank between 13 and 39 is too hard, maybe we can start with something easier. Can we figure out what’ll happen to 1 coin?
If you use the bag on a single coin twice, you end up with triple; that’s three gold pieces.
Because the bag always increases the number of gold coins, the blank must be between 1 and 3, so 2. It’s a start. What’s next?
Let’s think about a few other possible starting places.
We already know 2 becomes 3 and that lets us fill in the next blank as well. Now we’re getting somewhere!
We just need to extend this out to 13. Remember the other rule, though: when you put more coins in, you get more coins out. That means the numbers in every column must go in increasing order as well. In other words, because 6 coins become 9, it’s not possible for 4 coins to become 10. Nor could 4 become 5, since 3 becomes 6.
So 7 and 8 fill those blanks on the right of 4 and 5, which in turn gives the answer for two more blanks.
Knowing that the numbers go in increasing order in every column, the only choices for the remaining blanks are 19, 20, 22, and 23.
And look! We have our answer! There must be 22 gold coins in his hand.
“I’ll give you three guesses,” the little man begins to say.
“22 coins,” you respond.
“What?! How did you know?”
“I enjoy a good riddle,” you say. “Also, it’s on the back of your shirt.”
