## January 16, 2009

### Faulty coin problem

You are given twelve coins, of which eleven are genuine
and one is fake. All eleven genuine coins have the same weight
and the fake coin has a different weight. Find the fake in
three weighings.

SecretSquïrrel said...

Well, it's messy and complicated by the fact that we don't know whether the fake is light or heavy but bear with me.

Label the coins from A thru to L.

1. Weigh ABCD against EFGH.

a) If they balance then the fake coin is in IJKL; go to 2.

b) If ABCD is higher then either one of ABCD is fake and it's light OR one of EFGH is fake and it's heavy; go to 3.

c) If ABCD is lower then either one of ABCD is fake and it's heavy OR one of EFGH is fake and it's light; go to 4.

2. Weigh IJK against ABC (known to be not fake).

a) If they balance then L is fake. Weigh L against any other coin to see whether it's heavy or light.

b) If IJK is higher then the fake is one of IJK and it's light. Weigh I against J to see which is lighter. If neither then K is fake.

c) If IJK is lower then the fake is one of IJK and it's heavy. Weigh I against J to see which is heavier. If neither then K is fake.

3. IJKL are known to be not fake so weigh ABEF against CGIJ.

a) If they balance then the fake is either D (light) or H (heavy). Weigh DH against IJ. If DH is higher then D is the fake, otherwise H is the fake.

b) If ABEF is higher then either A or B are fake (and light) or G is fake (and heavy). Weigh A against G to see which it is. If neither then B is fake.

c) If ABEF is lower then either E or F are fake (and heavy) or C is fake (and light). Weigh E against C to see which it is. If neither then F is fake.

4. IJKL are known to be not fake so weigh ABEF against CGIJ.

a) If they balance then the fake is either D (heavy) or H (light). Weigh DH against IJ. If DH is higher then H is the fake, otherwise D is the fake.

b) If ABEF is higher then either E or F are fake (and light) or C is fake (and heavy). Weigh E against C to see which it is. If neither then F is fake.

c) If ABEF is lower then either A or B are fake (and heavy) or G is fake (and light). Weigh A against G to see which it is. If neither then B is fake.

Anonymous said...

1. 12--> 6|6
2.get 6(fault)--> 3|3
3.get 3(fault) --> 1|1 |1
either in first | second

Secret Squïrrel said...

@Anon

You can't go to your step 2 because you don't know which 6 are "faulty". The puzzle doesn't tell you whether the fake coin is lighter or heavier than the rest so you are unable to decide which of the 6 from your first weighing contains the fake coin. That's why you have to use a more complicated system.

brandon said...
This comment has been removed by the author.
Anonymous said...

hey secret squirrel,
i dun understand your (3rd and 4th) step B and C.
if 1 is heavy and 1 is light then which is the fake 1? the heavy 1 or the light 1?

Secret Squïrrel said...

You don't have 1 heavy and 1 light at the same time. Those are just the possibilities which you are determining from the balancing.

In step 3b, if ABEF is higher then either one of A / B is fake and lighter (since they're higher) or G is fake and heavier (since it is lower). We arrived here from step 1b so we already knew that if EF were fake, they would be lower and if C was fake it would be higher. In this case I and J are known to be true.

Anonymous said...

Take 6 hangs in the balance and 6 in the balance the other, then take a lighter and Nksmanm 3 is at stake and 3 in the other. Then take lighter and take 2 of whom were put up all alone in the balance. We have either to be equal and, therefore, that we take is not fake or be one of the scales lighter.