PORTER JOHNSON
ELEMENTARY MATHEMATICS SMILE A
21 MARCH 1995

WEIGHING PROBLEM

One is given a set of 12 coins, one of which is counterfeit and either lighter or heavier than the other 11. We are allowed to use a [crude] balance, which can only tell us whether the coins on the left pan weigh less than, the same as, or more than the coins on the right pan. Devise a scheme which involves only three different weighings to determine which coin is counterfeit, and whether it is lighter or heavier than the others.

Remark: To solve this problem one must make full use of the three possible results of weighing, varying which and how many coins are on each side, to develop the scheme. I have found a scheme which works, which I will describe here. It may not be unique.

Let us begin by numbering the coins 1 through 12. [It is important that the coins be distinguishable, in spite of having the same weight--perhaps the dates on them are all different.

Weighing #1: Put coins 1, 2, 3, 4 on the left pan and coins 5, 6, 7, 8 on the right pan.

• Case I: [0] The pan balances. We conclude that coins 1-8 all have the same mass, and that the counterfeit coin is either 9, 10, 11, or 12.

Weighing #2: Put coins 1, 2, 3 on the left pan and coins 9, 10, 11 on the right pan.

• Case A: [00] The pan balances. We conclude that coins 9, 10, 11 are OK, and that 12 is the bad coin.

Weighing #3: Put coin 1 on the left pan and coin 12 on the right pan. If left side is heavier [001] 12 is light, whereas if the left side is lighter [002] 12 heavy. [if the pans balance we should properly annihilate ourselves, to avoid a cataclysmic meltdown of the logical framework of the universe!]

• Case B: [01] The left side is heavy. We conclude that either 9, 10, or 11 is light.

Weighing #3: Put coin 9 on the left pan and coin 10 on the right pan. If they balance [010] coin 11 is light. If the left pan is heavier [011] coin 10 is light. If the left pan is lighter [012] coin 9 is light.

• Case C: [02] The left side is light. We conclude that either 9, 10, or 11 is heavy.

Weighing #3: Put coin 9 on the left pan and coin 10 on the right pan. If they balance [020] coin 11 is heavy. If the left pan is heavier [021] coin 9 is heavy. If the left pan is lighter [022] coin 10 is heavy.

• Case II: [1] The left side of the pan is heavier than the right side. We conclude that coins 9-12 are OK; and either one of the coins 1-4 is heavy, or one of the coins 5-8 is light.

Weighing #2: Put coins 1, 9, 10, 11, 12 on the left pan and coins 2, 3, 4, 5, 6 on the right pan.

• Case A: [10] The pan balances. We conclude that coins 1, 2, 3, 4, 5, 6 are OK, and either 7 or 8 is light.

Weighing #3: Put coin 1 on the left pan and coin 7 on the right pan. If left side is heavier [101] 7 is light, whereas if the pan balances [100] 8 is light. [if the left side is lighter [102] we again should undergo annihilation!]

• Case B: [11] The left side of the pan is heavier. We conclude either that either 1 is heavy, or that 5 or 6 are light.

Weighing #3: Put coin 5 on the left pan and coin 6 on the right pan. If left side is heavier [111] 6 is light, whereas if left side is lighter [112] 5 is light. If the pan balances [110] 1 is heavy.

• Case C: [12] The left side of the pan is lighter. We conclude that one of the coins 2, 3, 4 is heavy.

Weighing #3: Put coin 2 on the left pan and coin 3 on the right pan. If the left side is heavier [121] 2 is heavy, whereas if the left side is lighter [122] 3 is heavy. If the pan balances [120] 4 is heavy.

• Case III: [2] The left side of the pan is lighter than the right side. We conclude that either [a] one of the coins 1-4 is light, or [b] one of the coins 5-8 is heavy.

Weighing #2: Put coins 1, 9, 10, 11, 12 on the left pan and coins 2, 3, 4, 5, 6 on the right pan.

• Case A: [20] The pan balances. We conclude that coins 1, 2, 3, 4, 5, 6 are OK, and either 7 or 8 is heavy.

Weighing #3: Put coin 1 on the left pan and coin 7 on the right pan. If the left side is lighter [202] 7 is heavy, whereas if the pan balances [200] 8 is heavy. [if the left side is heavier [201] we again should undergo annihilation!]

• Case B: [21] The left side of the pan is heavier. We conclude either that either 1 is light, or that 5 or 6 are heavy.

Weighing #3: Put coin 5 on the left pan and coin 6 on the right pan. If the left side is heavier [211] 5 is heavy, whereas if the left side is lighter [212] 6 is heavy. If the pan balances [210] 1 is light.

• Case C: [22] The left side of the pan is lighter. We conclude that one of the coins 2, 3, 4 is light.

Weighing #3: Put coin 2 on the left pan and coin 3 on the right pan. If the left side is heavier [221] 3 is light, whereas if the left side is lighter [222] 2 is light. If the pan balances [220] 4 is light.

SUMMARY

 First Weighing Second Weighing Third Weighing *[000] harikari [100] 7 light [200] 8 heavy [001] 12 light [101] 8 light *[201] harikari [002] 12 heavy *[102] harikari [202] 7 heavy [010] 11 light [110] 1 heavy [210] 1 light [011] 10 light [111] 6 light [211] 5 heavy [012] 9 light [112] 5 light [212] 6 heavy [020] 11 heavy [120] 4 heavy [220] 4 light [021] 12 light [121] 2 heavy [221] 3 light [022] 10 heavy [122] 3 heavy [222] 2 light

Note: In three weighings with three types of outcomes, there are 27 possible total outcomes. We have distinguished 24 different configurations [each of 12 coins light or heavy], and there are three logical inconsistencies.