This One Weird Trick Will Get You Infinite Gold

The King’s Challenge

Imagine finding yourself in a dire predicament. A few years ago, the king decreed that your life would be forfeit unless you managed to triple the gold coins in his treasury. In your desperation, a strange little man appeared, offering a magical solution. He used a magical bag, placing handfuls of coins in and out while singing a peculiar rhyme. Miraculously, this act tripled the coins in the treasury, saving your life.

The Promise

Overwhelmed with gratitude and desperation, you promised him your firstborn child in exchange for his help. Fast forward to today, you’ve just given birth to a beautiful baby boy. The little man shows up to claim his prize. In your despair, you beg him not to take the baby. He softens and offers a challenge, “If you can guess my name—” you interrupt him, “Banach-Tarski?” noticing the name on his shirt. He is taken aback and quickly changes the challenge.

The New Challenge

The little man explains that his bag 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.”

The Mathematical Solution

The bag’s magic works just like a mathematical function. It’s convenient to use an arrow to denote the transformation. The challenge is to find out what goes in the blank between 13 and 39. The bag could be multiplying the number of coins by a certain number. If that’s the 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. But that’s not a whole number, and we don’t have bits of gold coins coming out of the bag. Something else is going on.

The Process of Elimination

If filling in the blank between 13 and 39 is too hard, 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?

Extending the Logic

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.

The Final Answer

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. The little man begins to say, “I’ll give you three guesses,” but you interrupt him, “22 coins,” you respond. He is shocked, “What?! How did you know?” “I enjoy a good riddle,” you say. “Also, it’s on the back of your shirt.”