How Can a Balancing Scale Help Solve the Mystery of the Counterfeit Coin?

[SchoolBoyDJ]

I just started physics this year, and we were assigned our first problem today. Before I continue I'd just like to stress that I am not looking for an answer. Just a hint, or something to help me out. Telling me that I'm on the right path would be helpful as well

Here's the problem:
-There are 12 coins, out of the 12 one is a counterfit.
-11 of the coins weigh the same, the counterfit weighs either more or less than a 'normal' coin.
-The only tool that may be used to solve this problem is a balencing scale, and it may be used only 3 times

So far I've come up with this:

Start off with 3 groups of 4 coins each.

Use Scale 1 Put two groups of 4 on each side of the scale, if they balence then you know that those 8 are all 'real' coins.

From here on I don't know what to do. I believe that you need to use groups of 4, but I'm not positive. What I do know is that you will need 2 procedures, 1 for if the first 2 groups balence, and one for if they don't.

Is my current thinking correct?

 

[radou]

Gee, I always hated such 'problems'... Okay.

Scaling 1. Put 4 coins on each side.
1a) equal weight => the different coin is among the 4 coins you're holding. Use Scaling 2 now and put 1 of these 4 coins on each side. 2a) equal weight => use Scaling 3 and put the two coins that are currently on the scale (and weigh equal) one the same side, no matter which one. Now, put the 2 coins on the other side. If that side goes down, you'll know the different coin is a heavier one, and you'll know which one it is because if felt (cheating?! ) heavier in your hands! If the other side goes up, you'll know the different one is a lighter one and you'll know which one it is.
2b) different weight => remember the position of the scale (which side is up, and which one is down). Now use scaling 3 to put the coins on the scale on the same side. Now, put the normal coins which you had in your hands on the other side. If the other side goes down, you'll know the different coin is a lighter one, and you'll know which one it is from scaling 2 (which side was up during scaling 2). If the other side goes up, you'll know the different coin is a heavier one, and you'll as well know which one it is from scaling 2.
1b) different weight => stuck here.. :zzz:

 

[e(ho0n3]

Coin-weighing problems can be modeled using decision trees. I won't go into depth about this but I will give you a hint: you need to label the coins.

 

[SchoolBoyDJ]

Gee, I always hated such 'problems'... Okay.

Scaling 1. Put 4 coins on each side.
1a) equal weight => the different coin is among the 4 coins you're holding. Use Scaling 2 now and put 1 of these 4 coins on each side. 2a) equal weight => use Scaling 3 and put the two coins that are currently on the scale (and weigh equal) one the same side, no matter which one. Now, put the 2 coins on the other side. If that side goes down, you'll know the different coin is a heavier one, and you'll know which one it is because if felt (cheating?! ) heavier in your hands! If the other side goes up, you'll know the different one is a lighter one and you'll know which one it is.
2b) different weight => remember the position of the scale (which side is up, and which one is down). Now use scaling 3 to put the coins on the scale on the same side. Now, put the normal coins which you had in your hands on the other side. If the other side goes down, you'll know the different coin is a lighter one, and you'll know which one it is from scaling 2 (which side was up during scaling 2). If the other side goes up, you'll know the different coin is a heavier one, and you'll as well know which one it is from scaling 2.
1b) different weight => stuck here.. :zzz:

This won't work because your proceedure needs to work. Everything needs to be written out, and work. No guessing, no weighing with hands etc :P

 

[SchoolBoyDJ]

Coin-weighing problems can be modeled using decision trees. I won't go into depth about this but I will give you a hint: you need to label the coins.

Ohhh...So there is a reason to my teachers madness. He had the 'coins' labeled with different letters. Thanks for pointing this out to me.

P.S. I was thinking more about this and I've gotten this far:

I had thought about the method the other person mentioned, and freaked out- I thought I had it all figured out, when I realized I hadn't found out if the fake coin was heavier or lighter than the others. What a bummer

 

[SchoolBoyDJ]

Well I've confirmed that you must start with groups of 4.

 

[fujiwara_sai]

hint: why don't you try taking some coins from here and there and weigh them together? Your grouping is correct but grouping sometimes is an unwanted restriction