Given a balance scale that is sure to break after X weighings, find an equation for the largest number of coins N, from which you can determine a fake coin that has the wrong weight if

A: You know whether the fake is lighter or heavier

B: You do not know whether the fake is lighter or heavier

(Assume only one of the N coins is fake)

Case 1: we know whether the anomalous coin is heavier or lighter. Then N=3^X

Case 2: we don't know. Then N = 3^(X-1)

For the first test, we divide the coins into 3 piles and compare two of the piles. After this first test, we are able to isolate the anomalous coin among a group of N/3 coins. We proceed similarly with this smaller group.

In case 1, after X tests, we will have isolated the anomalous coin.

In case 2, after X-1 tests we will have identified the anomalous coin as being one of two coins. We then compare one of these coins with a true coin. The result will tell us which of our two candidates is false.

Note: case 2 determination is only possible if X>1 

Posted by FrankM on 2008-02-16 13:07:19