You are given 11 real coins and one fake. the fake could be lighter or heavier than the real deal. you have a balance scale. In 3 weighs, determine which coin is the counterfeit and if it is heavier or lighter.

We have 12 coins. Each coin could be a fake and could be heavier or lighter than the real coins. Thus, we have 24 possibilities. We must determine the answer with only 3 weighs on a balance scale.

Setup

Break into 3 groups: A,B,C. Each coin is labeled 1-4 within each group (e.g., A1, A2, A3, A4, B1, …, C4)

Weight 1:

Weigh groups A and B. There are 3 cases:

i.) Scale tips towards group A

ii.) Scale tips towards group B

iii.) Scale is even

Without loss of generality, cases i and ii are identically, simply with different names. Thus, a solution for case i will hold true for case ii as well. Hence, we will solve two cases:

i) scale tips towards A

ii) scale is even

Case 1:

We now know that EITHER:

One of A1-4 is heavier OR one of B1-4 is lighter. Thus, we have 8 possibilities and 2 weighs left.

Case 1 Weigh 2:

Weigh A1-3 against C1-3. We have 3 sub-cases:

1. Scale tips towards A

2. Scale tips towards C

3. Scale is even

WLOG, cases 1 and 2 follow same logic. A solution to one will work for both. As such, I reduce cases 1 and 2 to case 1.

1.1 (and 1.2)

If the scale tips towards A, then one of A1-3 is the heavy counterfeit. Weigh A1 and A2. If it is even, A3 is the heavier counterfeit. Else, the scale will tip towards the heavier counterfeit. 1.1 and 1.2 QED

1.3

If the scale is even, then either A4 is heavier or B4 is lighter. Weight A4 and B4 on one side and C1 and C2 on the other. If the scale tips words C, then B4 is lighter, if not. A4 is heavier 1.3 QED

1 QED

Case 2:

We now know the counterfeit is in group C, thus any of the 4 could be heavier or lighter, we now have 8 possibilities and 2 weighs left.

Case 2 Weigh 2:

Weigh coins C1-C3 against A1-A3. We again have 3 cases.

1. Scale tips towards C

2. Scale tips towards A

3. Scale is even

2.1 (and 2.2) Weigh 3

WLOG cases 1 and 2 follow same logic.

If scale tips towards C, one of C 1-3 is heavier. Pick C1 and C2, then weigh to compare. If even C3 is heavier, if unbalanced the heavier is the counterfeit and it is heavier 2.1 and 2.2 QED

2.3 Weigh 3

If the scale is even, C4 is the counterfeit.

Weigh C4 with A1 to determine if it is heavier or lighter 2.3 QED

2 QED

FINAL QED