Wheighing Coins - looking for a legitimate answer?

Question

I will select a "best answer":

You have 12 coins which are identical in every way, except that one of them is counterfit in its weight, but you do not know which one it is or whether it is heavier or lighter than the others. (Assume, of course, that the difference is too small to tell just by holding it).

At your disposal you have a balance scale. Using this scale 3 times (no more), you must determine:

a) which coin is the fake
b) whether it is heavier or lighter than the others

Yes, it CAN be done - good luck!

Answer 1

You put 4x4 on the scale.

1) First consider the case of the scale being even. Then you have 8 coins that you know to be fair and 4 questionable. For your second use of the scale, measure 3 questionable coins against 3 fair ones.

If it comes out even, the remaining coin is fake, and you use the scale for the third time to find out whether it's heavier or lighter than any one of the 11 fair coins.

If it does not come out even than you have 3 coins, one of each is fake, but you already know if it's lighter or heavier. Let's assume it 's heavier. You can find the fake coin by weighing 2 of the 3 questionable coins against each other. If the scale is even, then it is the remaining coin, if it is not, then it is the heavier of the two. (*)

2) Suppose, the scale was not even. Let's number the coins C1 through C12, with the first 4 coins are heavier than coins 5-8. For your second use of the scale you put in: C1, C2, C3, C5 (left) against C4, C10, C11, C12 (right). If the left comes out heavier, then one of C1,C2,C3 is fake, and the fake is heavier, you already know how to solve that (*). If the scale is even, then one of C6, C7, C8 is fake, and the fake is lighter, so, again, you can use (*). Finally, if right is heavier, then either C4 is fake and heavy or C5 is fake and light. You can test either of them against any fair coin for your last use of the scale.

Answer 2

weigh three coins v three coins. If they don't balance then the fake lies in the coins chosen else it lies in the other six.

of the six including the fake weigh three against three known good coins narrowing the fake to a group of three

choose two of the fake group. If these balance then they are both good and you have found your fake. If they don't then you have no choice but to weigh a fourth time