# Coin Problem - Which is the counterfeit coin???

This is my Maths homework, and I really can’t figure out the answer.

I have 12 identical coins. One of them is counterfeit. The counterfeit coin may be lighter or heavier than a genuine coin. If I have a balance, how do I find the counterfeit coin within 3 weighings?

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this is an actual solution that works

number the coins 1-12

for weighing one, put coins 1-4 on one side of the scale and 5-8 on the other side

if the two sides in weighing one are equal, you know that one of the 9-12 coins are the counterfeit one. if this is the case, put coins 1-3(knowing that they are all real coins) on one side, and put coins 9-11 on the other side.

if they are equal, you know that coin 12 must be the counterfeit one, just weigh it alone against coin one, and you will know if it is heavier or lighter than the others.

if 9-11 are unequal to 1-3, you know if the counterfeit one is one of the 9-11 coins, and you know if it is heavier or lighter based on if 9-11 weighed more or less than 1-3. at this point, weigh coin 9 against coin 10, if they are equal, coin 11 is the fake, and you know if it is heavier or lighter based on the second weighing. if 9 and 10 are unequal, you know which one is fake based on knowing if the fake is heavier or lighter based also on the second weighing.

if the two sides in weighing one are not equal, then one of the coins 1-8 are fake (for convention, lets call 1-4 the heavier set). now take coins 2,3,6,7 and weigh them against 4,8,9,10.

if the two sets are equal, then 1 or 5 must be the fake, weigh coin 1 vs coin 2, if 1 weighs more, it is the fake, if they are equal, then 5 is the fake and it weighs less.

if 2,3,6,7 weighs more, then the fake is either 2,3,or 8. in this case, weigh 2 vs 3. if they are equal, then 8 is the fake and it weighs less. if either 2 or 3 weighs more, it is the heavier one and is fake.

if 4,8,9,10 weighs more, then the fake is either 4,6,or 7. in this case, weigh 6 vs 7. if they are equal, then 4 is the fake and it weighs more. if either 6 or 7 weighs less, it is the lighter one and is fake.

Get a magnet. If it is magnetized probably it is fake

I can do it if we assume that the counterfeit is lighter.

Weigh all 12 coins, six in each dish. The counterfeit will cause one of the sides to weigh less. The heavier six can be eliminated.

Take the six coins weigh three in each dish. Again, select the lightest side and eliminate the rest.

Out of the three remaining coins, choose any two. If one is lighter, it’s the counterfeit. If they’re the same, the one you left out is the counterfeit.

Well, you take all 12 coins and dump them on the scale. Then you’ll get a weight. Let’s just call this weight X.

Next, you take one coin (any one) and put it on the scale. Then weigh it. Call this coin cat. You’re not sure whether this coin is counterfiet or real right?

Third, drop any other coin (just not the same one you dropped in step 2) on the weighing scale. Call this coin Dog. Take note of the value.

All your weighing is done. You just need to ponder and calculate now

Look at the weights of the dog coin and the cat coin.

If they’re the same, then neither one of them is the counterfiet coin since there’s only ONE coin with a special weight. But, the good thing is that you know what one real coin weighs.

Using the weight of either the dog or cat coin (since they’re the same weight) and then multiply by 11, since you have 11 real coins. Then once you’ve got whatever weight that totals up to, subtract this from the total weight of all the 12 coins (which you measured in step 1).

This way, you can find the weight of that special coin, and you can easily know wwhat it’s weight is!

——————————————————————————

IF after step 2 and 3, dog and cat coins way different things, then you know that ONE of them is counterfiet.

Take the weight of dog and times 11. Then add the weight of the cat. This will make up all 12 coins and will have the totalled up weight. See whether this weight is equal to x which you worked out in step 1.

If it is more or less than X, then you know that the dog cannot be the real coin, since when you total up, it doesn’t work out. If the sum you got DOES equal to x, then you know that the dog coin is the weight of the real coin and the counterfiet is the cat coin.

Now, if the dog coin wasn’t right, you might want to check with the cat one. Take the cat coins weight and multiply by 11. Then add the weight of the dog coin.

If this weight is equals to X, then you know the counterfiet is the one weird dog coin.

Either the cat has 11 coins the same as it and one dog to add up to X, or the other way around.

Cheers.

seperate the 12 into three parts ..

that is 4 4 4

Weigh the the 1st 2 parts .

that is

weigh in balance

left: one part of 4

right :another part of 4

the third part is the rest…keep it..

now i can come to know that which one is heavier or lighter…

if both are equal ..then the counterfeit coin in the third part of 4…

like wise

4 4 4

2 2

1 1

First you should know counterfeit is lighter or heavier, suppose it is lighter:

Weighting #1:

12/2=6 => weight them in 2 different 6 coins’ group and find the lighter group.

Weighting #2:

6/2=3=> weight them in 2 different 3 coins’ group and find the lighter group.

Weighting #3:

Now you have 3 coins 1,2 and 3 . weight 1 and 2 if you find one lighter than other then it is counterfeit. if they are equal then 3rd number is counterfeit .