The Best Math or Logic Puzzle Ever

Updated on June 17, 2017
wilderness profile image

Dan is a family man, having raised two children, and has long been interested in the cultural, political, and social roots of our society.

Bet you can't solve this one!
Bet you can't solve this one! | Source

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 that give up quickly. While the answer is given below, I ask that you try, and try hard, to solve the riddle 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.

The Riddle

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 shown below with two pans hanging from a balancing arm. 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!

STOP HERE! Below is a picture of the scales to be used, followed by the solution. The 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.

Type of scales to be used in the riddle
Type of scales to be used in the riddle | Source

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.

Being 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, 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, 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, weight 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.

The left side is heavy.
The left side is heavy. | Source

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 heaver 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.

Winning poll - no fibbing now!

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Questions & Answers

    © 2011 Dan Harmon


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      • Kristen Howe profile image

        Kristen Howe 2 years ago from Northeast Ohio

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

      • jellygator profile image

        jellygator 5 years ago from USA

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

      • Judi Bee profile image

        Judith Hancock 5 years ago from UK

        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.

      • GoodLady profile image

        Penelope Hart 5 years ago from Rome, Italy

        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!

      • profile image

        Motown2Chitown 5 years ago

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

      • wilderness profile image

        Dan Harmon 5 years ago from Boise, Idaho

        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 profile image

        Brainy Bunny 5 years ago from Lehigh Valley, Pennsylvania

        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!

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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 profile image

        Debby Bruck 6 years ago

        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

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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 profile image

        Spencer Camus 6 years ago from United Kingdom

        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 ;-)

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

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

      • profile image

        sestasik 6 years ago

        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.

      • profile image

        Sean 6 years ago

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

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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.

      • profile image

        Sean 6 years ago

        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 :)

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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.

      • melbel profile image

        Melanie Palen 6 years ago from New Buffalo, Michigan

        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)

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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 profile image

        RedElf 6 years ago from Canada

        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

      • homesteadbound profile image

        Cindy Murdoch 6 years ago from Texas

        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.

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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 profile image

        Gordon Hamilton 6 years ago from Wishaw, Lanarkshire, United Kingdom

        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!

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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!

      • habee profile image

        Holle Abee 6 years ago from Georgia

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

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

        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 profile image

        dosters 6 years ago from Chicago

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

      • wilderness profile image

        Dan Harmon 6 years ago from Boise, Idaho

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

      • Allen Williams profile image

        Allen Williams 6 years ago from Pennsylvania

        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.