# 14 coins problem but you can't understand the scale

The 12 coins 12 coins problem but you can't understand the scale asks for is not the maximum possible, therefor this follow-up question:

You have a number of coins, one is fake but you don't know if it is lighter or heavier. You have a scale that gives you output but you don't understand it (let's say it is written in a foreign language that you don't understand).

For each possible answer (left, right, equal) there is only one output. Let's say you put 2 coins on the scale and the left is heavier, it A will be presented on the scale. But you see only A which you can't derive that it means that left is heavier because it is in some foreign language that you don't understand. But - A will be always if left is heavier, B for if right is heavier and C if it is equal, you'll just won't know it when you'll see the output.

Prove that using 4 weightings, the fake coin can be found in 14 but not in 15 coins.

If each coin is given by a different lower-case letter, you could use these tests:

Test 1: cefgjn vs bdhikm

Test 2: cehikl vs bdfgjn

Test 3: dfgikm vs bcehjl

Test 4: cdf vs gij

More coins can't be done because:

There are only 14 distinguishable results we can get from 4 weighings. In other words if A is the result of the first weighing, B is the result of the first weighing that isn't A (if any), and C is the result of the first weighing that isn't A or B (if any), then we can only receive AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC.

• @KrisVanBael f should give ABAA and h should give ABAC (which is your LRLE or RLRE). f should never give C/"equal" since it appears in all four tests. Commented Aug 21, 2023 at 18:41