# The Best Math or Logic Puzzle Ever

## The Riddle

Here is a logic puzzle that this author would bet you won't solve quickly. It is not for the faint of heart or those who give up quickly. While the answer is given below, I ask that you try—and try hard—to solve the riddle on your own before examining the answer. While the puzzle is based on both math and logic, it is primarily a logic riddle and should be treated as such.

## Ready?

You begin with 12 coins and the knowledge that one of these coins is counterfeit. The coins look identical, and the only difference between them all is that the counterfeit coin is of a different weight. You don't know whether it is lighter or heavier than the rest of the coins, just that it will weigh differently.

Your only tool is a set of old-fashioned scales, the type with two pans hanging from a balancing arm (see photo below). When more weight is placed on one pan than the other, that pan will sink while the lighter one will rise.

You have just three weighings to find the counterfeit coin. As you discover which one is counterfeit, you should also be able to tell if it is heavier than the rest or lighter.

Good luck!

### Your Objective

Your objective, of course, is to find the solution before reading how to solve the riddle. When you give up a week from now, come back and scroll slowly down to the next section; the first few sentences will give a valuable hint. Of course, if you figure it out, come back and let us all know in the poll at the bottom that it happened, or leave your online name in the comment section.

## Finding the Solution

Let's start by numbering the coins from 1 to 12, just so that they can be identified in this text. The actual solution doesn't require this, but it certainly makes it easier to describe the actions to be taken in solving the puzzle.

Begin by weighing coins 1, 2, 3, and 4 against 5, 6, 7, and 8.

**STOP HERE** if you are looking for a hint. That's it—weigh four coins against four coins. If you have given up and are looking for the answer, continue reading.

There are three possibilities. Let's call them case 1, case 2, and case 3.

- Case 1 is if both sides weigh the same; all the coins from 1 to 8 have the same weight, and none are counterfeit. Scroll down for the second weighing in case 1.
- Case 2 is if the left side, with coins 1, 2, 3, and 4, is heavier and causes that side sinks. This means that either one of these coins is heavy (and counterfeit)
**or**that one of the coins 5, 6, 7, or 8 is lighter and thus fake. Scroll down to see the second weighing for case 2. - Case 3 is the mirror image of case 2; the right side sinks. The solution for case 3 is analogous to case 2 and is left to the reader. The steps will be the same; you must simply visualize a different weight.

## Case 1: During Weighing 1, Both Sides Were the Same

The obvious conclusion is that one of coins 9, 10, 11, or 12 is counterfeit. For your second weighing, put coins 9, 10, and 11 on the left side and 1, 2, and 3 on the right side. There are, once again, three possibilities; case 4, where both sides are equal; case 5, where the left side goes down because it is heavier; and case 6, where the left side goes up because it is light. Bear in mind here that the right side contains *only* legitimate coins; if the left side goes down, it isn't because a light coin is on the right side; it is because a heavy coin is on the left.

In case 4, the solution is obvious; the bad coin is #12, the only one not proven to be of equal weight with all the others. For a third weighing, that coin can be measured against any other coin to determine if it is light or heavy.

For case 5, scroll down to find the third weighing if you haven't figured it out already.

Case 6 is again analogous to case 5 and is left to the reader to find.

## Case 2: During Weighing 1, the Left Side Is Heavier

This is the trickiest of all the possibilities. The author assumes you haven't figured it out—if you had, you wouldn't be here! Let's look at it.

For the second weighing, put coins 1, 2, 3, and 5 on the left pan and 9, 10, 11, and 4 on the right side. Remember that we already know that 9, 10, and 11 are all good coins.

There are (as always) three possibilities. Case 7 is that both sides weigh the same, and Case 8 is that the left side goes down. Case 9 is the right side goes down.

Case 7 tells us that the bad coin is number 6, 7, or 8 **and** that it is lighter than all the rest (remember weighing #1 where the side with those coins went up). All other coins have been proven to be of equal weight. For the third weighing, weigh 6 against 7; the one that goes up is the bad coin, while if they are equal, the final answer is coin 8.

Case 8 tells us that the bad coin is #1, 2, or 3 **and** that it is heavy. Were the problem in either coin 4 or 5, the scales would go the other way. For the third weighing, test coin 1 against coin 2. If one of them is heavy, it is bad, while if they are equal in weight, the bad coin is number 3.

Case 9 is that the right side of weighing 2 goes down. This can only happen if coin 4 is heavy or coin 5 is light. For the third weighing, test coin 4 against coin 1 (a known good coin). If unequal, coin 4 is bad and heavy; if equal, coin 5 is bad and light.

## Case 5: During Weighing 2, the Left Side Is Heavier

The conclusion to date is that the bad coin is number 9, 10, or 11 **and** that it is heavy. That side of the scales went down; it must be heavy.

As has already been seen, the solution to this scenario is to weigh #9 against #10; if one is heavier, that is the bad coin, while if they are equal, the bad coin is #11.

This finishes the solution to one of the very best math and logic puzzles or riddles. For those readers interested in mathematical problems, you might find Zeno's Paradox of interest. This was found in about 400 BC and has baffled thinkers ever since.

**© 2011 Dan Harmon**

## Comments

**Dan Harmon (author)** from Boise, Idaho on June 05, 2019:

I find no fault with your logic - you have found a different method. Congratulations!

**Jessica** on May 27, 2019:

My solution is a bit different but I think it still works:

Weigh 1, 2, 3, and 4 against 5, 6, 7, and 8

If they’re even, follow the first case above

If they’re uneven, weigh 1, 2, and 5 against 3, 4, and 6

If the scales go in the same direction, you know either 1, 2 or 6 is the counterfeit. Weigh 1 against 2. Same direction again, 1 is the counterfeit. Opposite direction, 2 is the counterfeit. Equal, 6 is the counterfeit

If the scales go in the opposite direction, you know 3, 4, or 5 is the counterfeit. Weigh 3 against 4. If they go in the same direction as the second 4 is the counterfeit. If they go in the opposite direction, 3 is the counterfeit. If they’re equal, 5 is the counterfeit.

If the scales are equal on the second weighing, you know either 7 or 8 is the counterfeit. Weigh them against each other to see which aligns with the initial direction of the 5, 6, 7, 8 weight

To keep it short I left out heavy/light, but you can tell which the counterfeit is by writing out all the possible solutions to these steps!

**Kristen Howe** from Northeast Ohio on April 24, 2015:

Dan, this was a good hub for math puzzles. It does boggle your mind. Voted up for interesting!

**jellygator** from USA on October 09, 2012:

I've seen this before, but I didn't remember it until I read your first instructions. Fun stuff!

**Judi Brown** from UK on October 09, 2012:

I'm useless at this kind of thing! Great fun though - will share it for sure - could be a good end of term challenge for the kids too.

**Penelope Hart** from Rome, Italy on October 09, 2012:

Sending this to two members of my huge family who adore puzzles. I'm sure Im dyslexic because i can NOT do anything so clever - though this hub, I do admire!

**Motown2Chitown** on October 09, 2012:

I feel like a genius! :) This was really neat and I'm about to share it.

**Dan Harmon (author)** from Boise, Idaho on April 20, 2012:

I like these kind of things, too, and enjoy working them. This one impressed me as it took a long time to work out all the detains - just wanted to share it with others.

**Brainy Bunny** from Lehigh Valley, Pennsylvania on April 19, 2012:

I love logic puzzles, but I usually don't actually take the time to figure them out. For this one, I immediately knew the general procedure to follow, but I didn't have the patience to work it all out. But it was a fun hub!

**Dan Harmon (author)** from Boise, Idaho on December 24, 2011:

Glad you enjoyed it, Debby - I have always liked this puzzle. And that's exactly how I finally figured it out - sitting down with a bunch of coins.

**Debby Bruck** on December 21, 2011:

Hi Wilderness ~ This is a super puzzle. I took out my coins and got some inkling how to get started. Your explanation really sorted it out well. First thing I thought of when chose this hub to read was 'Car Talk' which I also love. Blessings, Debby

**Dan Harmon (author)** from Boise, Idaho on December 05, 2011:

LOL It's good for you! I, too, like these kind of puzzles and it probably actually is good for us to stretch those mental muscles. Glad you liked it!

**Spencer Camus** from United Kingdom on December 05, 2011:

Aaargghhh, I hate you!! Only joking. I find these type of puzzles frustrating but can't help but sit and figure it out. I managed after reading the hint, and got it very quickly after that. What a great hub, and very well explained. Voted up and funny...there was no 'frustrating' option ;-)

**Dan Harmon (author)** from Boise, Idaho on December 02, 2011:

That would be fun, wouldn't it? Actually have students with the coins and scales weighing it out each time.

**sestasik** on December 02, 2011:

Great Hub, and share worthy. Would be an awesome exercise in a classroom setting, especially if you could actually supply the coins and scales. Voted up and shared.

**Sean** on November 27, 2011:

Ah, I thought the counterfeit was lighter! Damn. thanks :)

**Dan Harmon (author)** from Boise, Idaho on November 25, 2011:

Sorry, it won't work. If you weigh one group of six against another, one group will be heavier than the other which you knew would happen before ever weighing them.

What you don't know is which one contains the counterfeit; you don't know if the bad coin is heavy or light. Just that it is different. This is explained in the second paragraph and is what makes the puzzle so difficult.

Good try, though, and a common error in solving the puzzle.

**Sean** on November 25, 2011:

Cool riddle! I've got another answer:

1. Split the 12 into two groups of 6 and weigh these two groups. Now you know which group of 6 the counterfeit is in.

2. Split the group of 6 that contains the counterfeit into two groups of 3. Now weigh these two groups of 3.

3. Now you've narrowed the counterfeit down to one of three coins. Choose any two of these coins to weigh.

If one is lighter, it's the counterfeit.

If they're the same, the one not weighed is counterfeit.

Voila! I think :)

**Dan Harmon (author)** from Boise, Idaho on October 30, 2011:

They are fun to do, aren't they. I had seen this as well, decades ago and using something other than coins, and remembered just enough to set it up. I then had to find the answer, and it wasn't easy!

Thanks for the compliment, melbel - it means a little more coming from you.

**Melanie Palen** from Midwest, USA on October 30, 2011:

I have seen this puzzle before and I admit that I already knew the answer. However, when I first found the answer, it was ALSO from peeking. These types of questions are often used in interviews for those who apply at places like Google. I am complete rubbish at figuring these out. :P I think I've only figured out one of these types of puzzles in my life.

Great hub, by the way! I love to attempt to solve these kinds of things (even though I eventually give up and peek. :P)

**Dan Harmon (author)** from Boise, Idaho on October 30, 2011:

If you got by the first hint without looking at it you didn't do badly. It's that second weighing, though, with the funny side to side switch for two coins, that throws most people. They forget that at that point they have four known good coins to work with and to simply swap two of the 8 coins on the scales doesn't occur to them either.

**RedElf** from Canada on October 30, 2011:

Very clever puzzle - I got as far as finding the bad coin (#12 in your solution) if both sides weighed the same both times. I needed some help to sort it out if one side was light or heavy on the second weighing, but didn't need the first hint. Thanks for the work-out! ...and Zeno's Paradox isn't really a paradox, is it? I will go read your hub and find out. :D

**Cindy Murdoch** from Texas on October 28, 2011:

I don't know why, but I came up with the answer right away. It just seemed logical. This is a great hub. I'm sending it to my husband for him to look at.

**Dan Harmon (author)** from Boise, Idaho on October 27, 2011:

LOL It's tough to keep track of, isn't it? It's that case #2 that throws most people - it just isn't very intuitive.

**Gordon Hamilton** from Wishaw, Lanarkshire, United Kingdom on October 27, 2011:

Your poll doesn't allow the option, "I had to read all of the solution several times before it hit me!" :) Although maths and physics were always my two strongest subjects and should give me a good grounding/chance in problems of this type, they never have - I'm hopeless with these conundrums, however many I try to solve. I don't honestly believe I would have figured this out in ten years of constant pondering. Good question!

**Dan Harmon (author)** from Boise, Idaho on October 27, 2011:

It's a tough one. My brother (a statistician and programmer) and I played with a dozen coins of different values (to differentiate them) one day for several hours and never did come up with the answer. It took me another couple of days to find it.

It will definitely give those brain cells a workout; at my age I need that!

**Holle Abee** from Georgia on October 27, 2011:

Ummm...huh?? I have no left brain. lol. Great riddle, though. Voted up!

**Dan Harmon (author)** from Boise, Idaho on October 26, 2011:

Good idea - I may work on that and see if I can come with something. I know the solution is a little hard to follow; there are too many possibilities to make it simple. Glad you enjoyed it.

**dosters** from Chicago on October 26, 2011:

You should include a flowchart at the bottom. It would make the explanation easier to follow. Great puzzle.

**Dan Harmon (author)** from Boise, Idaho on October 25, 2011:

Wow! You're good! It took me a couple of days to come with the answer.

**Allen Williams** from Pennsylvania on October 25, 2011:

I actually figured it out in a couple of minutes, but that is only because I have been doing puzzles like this for many years. Since it was a good hub with a good problem, I voted it up. Thanks, I like stuff like this.