There are 12 boxes each containing a coin. There are a total of 11 pennies and 1 other coin in the boxes. You have a balance scale for comparing the weight of the boxes in which you can put any number of boxes on each side. You have at most, three weighings. In these three weighings you must find out which of the 12 boxes has the coin that is not a penny.
Balance any 4 coins (and call them A) with any other 4 coins (and call them B).
Case 1. - The coins balance. You now have a stock of 8 coins (A and B) you know are ok. Weigh 3 good coins again 3 unknowns - if they balance then you have identified the coin and can use the last balance to determine if it is heavy or light. If they don't balance you know if it is heavy or light. Weigh 2 of the 3 unknown coins, and they will balance (identifying the unknown coin) or not.
Case 2 - Left hand balance (A) is heavier. You have a stock of 4 known coins (C) and 8 unknown coins (A and B). Take 2 coins from group A (the left hand side) and 2 from group B (the right hand side) and balance these with 1 coin from A, 1 from B and 2 from C. If they balance, then the 4th coin from A or B is the unknown coin, and you can balance with a good coin to determine which. If A1, A2, B1, B2 are lighter than A3, B3, C1, C2 then either B1, B2 are lighter or A3 is heavier (and can be determined by a single balance) If A1, A2, B1, B2 are heavier than A3, B3, C1, C2 then either A1, A2 are heavier or B3 is lighter (and can be determined by a single balance)
Case 3 - Left hand balance (A) is lighter.
As case 2, but swap all heaver / lighter.
An alternative solution is often proposed (This only works if you know in advance that the none-penny is heavier. As this is not stated in the puzzle it will not work in all cases)
1. Divide the coins into two groups, each containing six coins. Weigh the two groups and set aside the lighter group.
2. Divide the heavier group into two stocks, each containing three coins. Weigh these two stocks, setting aside the lighter stock.
3. Weigh two coins from the heavier stock against each other. If either coin is heavier, then the heavier coin is not a penny. If the coins weigh the same amount, then the third coin in that stock is not a penny.