A Strategy For Solving The Forged Coin Problem
Edward J Robinson [Collins HS]
28 March 2000

There are twelve coins, one of which is forged. The forged coin is either lighter or heavier than the others. By using at most three weighings on a balance scale, find the forged coin and determine whether it is heavier or lighter than the other coins.

Divide the twelve coins into three groups of four, groups A,B,C.

First weighing

Place group A on the left and place group B on the right. Set group C aside. If the balance is equal then the forged coin is in group C. If the balance tends to the left, then the forged coin is either in group A or B. If the forged coin is in group A, then it is heavier; if the forged coin is in group B, then it is lighter. If the balance tends to the right, then the forged coin is either in group A or B. If the forged coin is in group A, then it is lighter; if the forged coin is in group B then it is heavier.

• Result of first weighing- Balance equal

Second weighing

Place any three coins from group A on the left and any three coins from group C on the right.

Result of Second weighing

• If the balance is equal, the coin set aside from group C is forged and you can weigh it against any of the other coins for the third weighing to determine whether it is heavier or lighter. Place the forged coin on the left. If the balance tends to the left then the forged coin is heavier. If the balance tends to the right then the forged coin is lighter.

• If the balance tends to the left. the forged heavier coin is on the left. Take off the three coins from the right side. Place on the right any one of the three coins that are on the left. Put aside the third coin from the left. If balance is equal then the third coin set aside is forged and heavier. If balance tends to the left then the coin on the left is forged and heavier. If the balance tends to the right then the coin on the right is forged and heavier.

• If the balance tends to the right, then the forged lighter coin is on the left. Take off the three coins from the right side. Place on the right any of the three coins that are on the left. Put aside the third coin from the left. If balance is equal then the third coin set aside is forged and lighter. If the balance tends to the left then the coin on the right is forged and lighter. If the balance tends to the right then the coin on the left is forged and lighter.

• Result of First weighing- Balance tends to the left

Second Weighing

Switch the positions of the two coins that are on the far left on both the left side and the right side. On the right side, replace the non- switched coins, (group U) with three coins from group C.

Results of Second weighing

• If the balance is equal, then the forged coin is either one of the three replaced coins (group U), which can only be a heavier coin. You can weigh any two of the coins. If the balance is equal then coin not weighed is heavier and forged. If the balance tends to the right then the forged heavier coin is on the right side. If balance tends to the left, then the forged heavier coin is on the left.

• If the balance tends to the left then the lighter forged coin is on the right side but not the farthest left coin. The farthest left coin on the right side cannot be a lighter coin because it was on the left when the balance tended to the left after the first weighing. The farthest left coin on the left side cannot be a heavier coin because it was on the right when the balance tended to the left after the first weighing. Take off the coins on the left side. Take off the farthest left coin on the right side. Move one coin from the left side to the right side. Place one coin aside from the right side. If the balance is equal, then the coin placed aside is lighter and forged. If the balance tends to the right then the forged lighter coin is on the left side. If balance tends to the left, then the forged lighter coin is on the right.

• If the balance tends to the right then the forged coin is either the farthest left coin on the left side or the farthest left coin on the right side. Leave these two farthest left coins on the scale and take the other coins off the scale. Weigh either coin against a known regular coin to determine the forged heavier or lighter coin. For example, replace the coin on the right side with a known regular coin. If the balance is equal, then the coin placed aside is forged and heavier. If the balance tends to the right, then the forged lighter coin is on the left side.

• Result of First weighing- Balance tends to the right

Second Weighing

Switch the positions of the two coins that are on the far left on both the left side and the right side. On the right side, replace the non- switched coins, (group U) with three coins from group C.

Results of Second weighing

• If the balance is equal, then the forged coin is either one of the three replaced coins (group U), which can only be a lighter coin. You can weigh any two of the coins in group U. If the balance is equal then coin not weighed is lighter and forged. If the balance tends to the right then the forged lighter coin is on the right side. If balance tends to the left, then the forged lighter coin is on the left.

• If the balance tends to the right, then the heavier forged coin is on the right side but not the farthest left coin. The farthest left coin on the right side cannot be a heavier coin because it was on the left when the balance tended to the right after the first weighing. The farthest left coin on the left side cannot be a lighter coin because it was on the right when the balance tended to the right after the first weighing. Take off the coins on the left side. Take off the farthest left coin on the right side. Move one coin from the left side to the right side. Place one coin aside from the right side. If the balance is equal, then the coin placed aside is heavier and forged. If the balance tends to the right then the forged heavier coin is on the right side. If the balance tends to the left, then the forged heavier coin is on the left side.

• If the balance tends to the left then the forged coin is either the farthest left coin on the left side or the farthest left coin on the right side. Leave these two farthest left coins on the scale and take the other coins off the scale. Weigh either coin against a known regular coin to determine the forged heavier or lighter coin. For example, replace the coin on the right side with a known regular coin. If the balance is equal, then the coin placed aside is forged and lighter. If the balance tends to the left, then the forged heavier coin is on the left side.