Find one fake coin out of 14 in 3 scales

Question

There is a pile with 14 coins. One of them is fake and has a different weight than the others, which are the same.

Using a balance scale, find the fake coin with no more than 3 scales.

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Based on the wiki article it is believed that no more than 13 coins can be tested with 3 scales, but I found a solution for 14 coins, and I wish to share it with you in my answer below.

Answer 1

Take two coins and slice them in half with equal weight.

• Scale 1: $3+0.5 + 0.5$ vs. $4$ -> Take 2 distinct half coins with 3 other coins, let's call it "pile A", and weigh them against 4 other coins. Let's call it "pile B". And we are leaving 5 coins on the side.
• If the scale is not balanced, it means that the 5 coins we didn't scale are not fake.
  • Scale 2: $2 + 3B$ vs. $1 + 4R$ -> Take 2 coins from pile A with 3 coins from pile B and weight them against 1 coin from pile B and 4 real coins(those we didn't weight and we now know are not fake)
  • There are three possible outcomes: the scale is flipped, the scale is balanced, or it didn't change. In all those cases, we left out with three coins where we know the relative weight of a fake coin.
  • Scale 3: $1$ vs. $1$: Weight two of the coins in question, one against the other, leaving one aside. In the case of half coins, compare half and half coins. If the scale is balanced, then the fake coin is the one we put aside. Otherwise, we should know who is the fake coin from the scale result, as in this case, we know the relative weight of the fake coin.
• In case the scale is balanced, it means that the fake coin is among the 5 coins we didn't weight
  • Scale 2: $2$ vs. $1 + 1R$ -> Take two coins, let's call it "pile A" and scale them against 1 coin and 1 real coin, let's call it "pile B".
    • If the scale is balanced, then the fake coin is one of the 2 we didn't scale. Scale one of them with a real coin to find out who is the fake coin.
    • Scale 3: $1 + 1B$ vs. $2R$ -> If the scale is not balanced, then scale one coin from pile A with one coin from pile B against two real coins. There could be three outcomes: The scale is balanced; the scale is flipped; the scale didn't change. That correlates with the 3 coins we have been testing to find the fake coin.