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