Arrr! Gold coins!

Probability Level 3

There are 4 piles of gold. 1, 6, 19, 96 gold coins in each respectively. In one turn you are able to take 1 coin from 3 chests (doesn't matter which ones) and place it into another chest. Will you be able to put all the coins in one chest?

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Zyberg Nee by Zyberg NEE , 21, Lithuania

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1 solution

Zyberg Nee
Jun 14, 2016

Relevant wiki: Invariant Principle

1 ≡ 1 m o d 4 1 \equiv 1 \mod 4

6 ≡ 2 m o d 4 6 \equiv 2 \mod 4

19 ≡ 3 m o d 4 19 \equiv 3 \mod 4

96 ≡ 0 m o d 4 96 \equiv 0 \mod 4

All reminders are distinct. After one turn all of them will shift, but remain distinct.

For example, I took 2nd, 3rd and 4th piles and placed coins into the first pile, then I would get:

4 ≡ 0 m o d 4 4 \equiv 0 \mod 4

5 ≡ 1 m o d 4 5 \equiv 1 \mod 4

18 ≡ 2 m o d 4 18 \equiv 2 \mod 4

95 ≡ 3 m o d 4 95 \equiv 3 \mod 4

To have all the coins in one pile it would require having remainders to be 0, 0, 0, 2. It is not possible.

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