Can you solve the "infinite gold" math puzzle

Originally published at: Can you solve the "infinite gold" math puzzle | Boing Boing

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No, I cannot. I didn’t need to watch the video either. Just saw “math” and “puzzle” and instinctively knew my answer.

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Meanwhile a mathematician would answer “yes, I’m sure I can”, and then leave the details an exercise to the reader.

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I’m disappointed. I thought they were going to come up with a simple equation to calculate the second value for any starting amount of coins. I want my gold back!

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n → n + 3ⁱ if 3ⁱ ≤ n < 2∙3ⁱ for some i
n → 3(n - 3ⁱ) if 2∙3ⁱ ≤ n < 3∙3ⁱ for some i

Yeah, I was expecting them to give an equation too, I worked backwards from the examples to the equation.

It’ll be a lot easier to see what’s going on if the numbers are written out in base 3:
If the leading digit is 1 in base 3 change the leading digit to 2.
If the leading digit is 2 in base 3 change the leading digit to 1 and add a 0 to the end.

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Very sweet solution.
We worked it out by hand up to 50 or so and noticed that the jumps were always by 1 or 3, but didn’t quite get to this tight an answer.

I guess that means there are versions of the problem where you apply f() k-1 times and the number gets multiplied by k. Increment the leading digit (in base k) unless it’s k-1, in which case change it to 1 and append a 0.

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That’s wonderful.

If I had a dollar for every time I’ve said that…

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I don’t like how the puzzle is worded. He’s like “more come out” and I’m like “okay do you just mean each time you put in an amount of coins, an amount of coins comes out again because if any come out, that’s increases the amount that had come out ever previously - or do you mean the amount of coins that comes out is strictly greater than the amount put in?” also “how many coins do you have” - do you mean in your hand? Does the bag also contain the coins you put in it in the first place and are you asking about them too? I have so many questions. It made me think of the “what kind of swallow - African or European?”

Not a mathematician, so please forgive the probably stupid question, but I take it those two are mutually exclusive? E.g., if the first is true for some value of i, there is no value of i for which the second is true?

Also, one or the other must be true, for exactly one value of i — is that correct?

ETA: please ignore, just proven both to myself on paper. Obvious now. D’oh!

Can someone explain how this is linked to Banach-Tarsky? I roughly get that this theorem is about making copies from one thing, but is there any other mathematical linkage?

Nah. One of the weird things about Banach-Tarski is that it’s a proof that there exists a way to double a sphere, not, an actual rule that does so. Moreover, one can prove (Solovay 1970) that there will be no rule. Whereas werdnagreb has given us a rule.
So, it’s just a joke reference.

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