hard 12 coin problem!!!?

Question

i have 12 coins and 11 of these are identical, and the 12th is a fake coin. it weighs either slightly more or slightly less than the other 11, but looks exactly the same. is it possible to find the fake coin in 3 weighings, using scales, and how do u do it?

Answer 1

Weigh 4 against 4.
-If the scale balances, you know the fake is among the remaing 4
Weigh 3 unknowns against 3 knowns.
--If it balances, the remaining unknown is fake, weigh against a known, to see it it is too heavy or too light.
If it doesn't balance (say the unknowns are too heavy) the fake is among those three. Compare two of the three the heavy one is fake, if they balance the last one is fake (and too heavy)

That was the easy case. Say when you weighed 4 v 4 earlier, that one side was too heavy. You know the 4 untouched are real. Weigh HHL vs HHL. Where H=heavy, L=light) (You have 2 L coins left)
-If they balance, they are all real and the fake is one of the L's left, compare 'em.
-If one side tips, either one of the heavy coins is fake or the light one the other side is fake. Compare the heavy coins, the heavy one is fake, otherwise if they balance the light coin on the other side was fake.

Answer 2

A minimum of 4 steps are required if you do not know for sure if the fake coin is heavier or lighter. However, if you do know if it is either heavier or light, it would be easier, and would only require 3 steps.

Divide the 12 coins into 3 groups of 4 coins each, A, B and C.
1) Balance A and B.
If they balance, then fake coin is in C. (Go to 3)
If they don't balance, then the fake coin is in group A or B. (Go to 2)

2) Balance A and C.
If they balance, the fake coin is in B. (Go to 3)
If they don't balance, the fake coin is in A. (Go to 3)

Of the 4 coins in this group, name them D, E, F and G.
3) Balance D and E.
If they balance, the fake coin must be in either F or G. (Go to 4)
If they do not balance, the fake coin must be in D or E. (Go to 5)

4) Balance D (or E) with F.
If it balances, then the fake one is G.
If it does not balance, the fake on is F.

5) Balance D with F.
If it balances, then the fake one is E.
If it does not balance, the fake on is D.

Answer 3

kemmy's right, if you don't know if the fake is heavier or lighter you need 4 steps.

1) Split the 12 into 3 piles of 4, A, B and C. Weight A against B. You may get lucky and they're identical, in which case you can skip step 2 as the fake is in pile C.

2) If they weigh different, you need to discover if the fake is heavier or lighter, so weigh A against C. If these are identical, the fake is in B, if not then the fake is in A.

3) take the fake pile and split it into 2 piles, we'll call D and E. Weigh D against E. If you know whether the fake is heavier or lighter, you will know which of these piles the fake is in, and simply weigh the 2 coins in this group against each other to find the fake.

4) If you don't know the weight difference, swap one coin from D and one from E and reweigh. you can then determine the fake from these results.

Answer 4

To solve you need a "balance" that compares two weights.

I attack this type of problem using matrix methods - creating a linear model based on a three level measurement (heavy/same/light). If we can make three independent measurements we will maximize the information gained and (as it turns out) have enough information to answer the question.

Answer 5

its always possible its just unlikely that you will get lucky in a three out of tweleve sampling