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.