**SOLUTION TO THE 12 COIN PROBLEM**

**IF** one had 12 seemingly identical coins, with 11 being of the exact same weight and 1 being either heavier or lighter than the other 11,

**THEN** using only a balance, not a scale, and with only 3 measurements allowed, how could one determine which of these 12 seemingly identical coins was different and whether it were heavier or lighter than the other 11?

**Here is the solution:**

**Step 1**: Balance coins 1, 2, 3 & 4 against coins 5, 6, 7 & 8: **(1,2,3,4) | (5,6,7,8)**

**The first possible outcome from Step 1:**

If **Step 1** yields Heavy on Left, then either one of 1, 2, 3 or 4 is Heavy or one of 5, 6, 7 or 8 is Light.

**Step 2**: Balance 1, 5, & 6 against 2, 7, & 9: **(1,5,6) | (2,7,9)**

If **Step 2** yields Heavy on the Left, then either 1 is Heavy or 7 is Light.

**Step 3**: Balance 1 & 7 against 9 & 10: **(1,7) | (9,10)
**Heavy on Left: 1 is Heavy

Light on Left: 7 is Light

Equal is not possible

If **Step 2** yields Light on Left, then either 2 is Heavy or 5 or 6 is Light.

**Step 3**: Balance 5 against 6: **(5) | (6)
**Heavy on Left: 6 is Light

Light on Left: 5 is Light

Equal: 2 is Heavy

If **Step 2** yields Equal, then either 3 or 4 is Heavy or 8 is Light.

**Step 3**: Balance 3 against 4: **(3) | (4)
**Heavy on Left: 3 is Heavy

Light on Left: 4 is Heavy

Equal: 8 is Light

**Back to Step 1 – The second possible outcome from Step 1:**

**Step 1**: Again, balancing 1, 2, 3 & 4 against 5, 6, 7 & 8: **(1,2,3,4) | (5,6,7,8)**

If **Step 1** yields Light on Left, then either one of 1, 2, 3 or 4 is Light or one of 5, 6, 7 or 8 is Heavy.

**Step 2**: Balance 1, 5, & 6 against 2, 7, & 9: **(1,5,6) | (2,7,9)**

If **Step 2** yields Heavy on Left, then either 5 or 6 is Heavy or 2 is Light.

**Step 3**: Balance 5 against 6: **(5) | (6)
**Heavy on Left: 5 is Heavy

Light on Left: 6 is Heavy

Equal: 2 is Light

If **Step 2** yields Light on Left, then either 1 is Light or 7 is Heavy.

**Step 3**: Balance 1 & 7 against 9 & 10: **(1,7) | (9,10)
**Heavy on Left: 1 is Light

Light on Left: 7 is Heavy

Equal is not possible

If **Step 2** yields Equal then either 3 or 4 is Light or 8 is Heavy.

**Step 3**: Balance 3 against 4: **(3) | (4)
**Heavy on Left: 4 is Light

Light on Left: 3 is Light

Equal: 8 is Heavy

**Back to Step 1 – The last possible outcome from Step 1:**

**Step 1**: Again, balancing 1, 2, 3 & 4 against 5, 6, 7 & 8: **(1,2,3,4) | (5,6,7,8)**

If **Step 1** yields Equal Left to Right balance, then one of 9, 10, 11 or 12 is either Light or Heavy.

**Step 2**: Balance 9 & 10 against 11 & 1: **(9,10) | (11,1)**

If **Step 2** yields Heavy on Left then either 9 or 10 is Heavy or 11 is Light.

**Step 3**: Balance 9 against 10: **(9) | (10)
**Heavy on Left: 9 is Heavy

Light on Left: 10 is Heavy

Equal: 11 is Light

If **Step 2** yields Light on Left then either 9 or 10 is Light or 11 is Heavy.

**Step 3**: Balance 9 against 10: **(9) | (10)
**Heavy on Left: 10 is Light

Light on Left: 9 is Light

Equal: 11 is Heavy

If **Step 2** yields Equal Left Right balance, then 12 is either Heavy or Light.

**Step 3**: Balance 12 against 1: **(12) | (1)
**Heavy on Left: 12 is Heavy

Light on Left: 12 is Light

Equal is not possible