Puzzle Contest #7 – Twing’s Homework
Posted on 07. Mar, 2010 by Chris Matney in Front Page Posts, Puzzle
Matt Twing sent in a classic riddle which I thought I would share with you. I suspect it might be a homework problem for his logic class, as Matt is a high school senior and sneaky smart.
You have 12 marbles, one of them is either heavier or lighter than the rest. In three weighs on a balance scale, you must determine which is the odd marble. How do you do this?
6 Responses to “Puzzle Contest #7 – Twing’s Homework”
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This is a tricky problem. Since I can’t solve it, I’m going to cheat.
We have 3 weighings. We’re going to use Weigh #1 by stacking 6 marbles on each side. But, you can’t just go tossing marbles willy-nilly onto the scales. So, we’re going to place them slowly, two at a time, one on each side, on the way to our first weighing of 6 and 6.
However, at some point when we’re placing the marbles two by two, the scale will unbalance. Oops! Now we know that one of the two most recently added marbles is the culprit (you can continue to stack until you have 6 per side if it makes you feel better about bending the rules).
With our next weighing, we weigh either marble against any previous marble. If it’s unbalanced, you have the culprit (we know that all the previous marbles were balanced since we stacked them in opposite pairs).
If those two balance, then weigh the other marble against any normal marble. It will either be heavier or lighter, and again you have the outlier.
Some would argue that slowly placing the marbles opposite each other constitutes weighing them. I say it’s just not my fault if I’m a slow marble stacker on my way to weighing 6 vs 6 one time.
You’re right, Jeff. This is more devious than I thought. Let me think about this one.
Tim,
I don’t think your solution works for all cases. You start by creating 4 groups of 3 marbles, A B C D. Say the odd marble is in group D but you don’t know if it’s heavier or lighter. You start by weighing 2 groups against each other. If you start with A and B, they will balance. For the second weighing, you can’t weigh C against D because not knowing if the odd marble is heavier or lighter, you would still end up with 6 possibilities. As you suggested, you can weigh a group you know to be normal against an unknown group. So lets say you weigh A against C. They will balance, so you know the odd marble is one of the 3 in group D. Now you put 2 marbles from D on the scale. If the scale balances, you know the odd marble is the one you didn’t weigh. However, if the scale doesn’t balance, you don’t know which marble is the odd one because you still don’t know if the odd one is heavier or lighter.
Very close.
Jeff, you were close. You need to use four groups not three.
Take the twelve marbles, divide them into four groups of three (A, B, C, D). For the sake of simple explanation, let’s say group C has a heavier marble.
Put one group of 3 on each side, if they balance, then these groups do not contain the marble of interest. If they don’t balance, then one group contains the marble. Note carefully which group is heavier and which lighter. If we chose to weigh A and C, the side with C would be heavier.
Take a normal group of 3 (i.e. either B or D) and weigh them against either other group (A or C). If you chose A, then the scales balance (hence not lighter) and you know group C has your unbalanced marble. If you chose C, then the scales don’t balance and you know C has unbalanced marble. You also know C is heavier than the normal group.
From C, then chose two marbles at random to weigh, if they balance, the one you set aside is the unusual one. If they don’t then the unusual marble is the one that is either heavier or lighter, depending on the results of the second weighing. In this case, it is the heavier marble.
Jon’s solution works if you know that the odd marble is heavier (or lighter). If you don’t know the relative weight of the odd marble it’s more complicated. You can still narrow it down to 6 marbles after the first weighing by putting 3 marbles on each side of the scale. If the scales balance, the odd marble must be in the group of 6 you didn’t weigh. If the scales are unbalanced, the odd marble is in the 6 you did weigh, but since you don’t know if it is lighter or heavier, you don’t know which side it’s on so you can only narrow it down to all 6. I couldn’t figure out a way to determine the one odd marble from a group of 6 with 2 trials remaining not knowing if the marble is heavier or lighter, so I went to plan B.
For the first weighing, divide the 12 marbles into 3 groups of 4 marbles. Put one group on each side of the scale and set aside the remaining 4 marbles. If the scale balances, you know that the odd marble is in the group of 4 that you didn’t weigh. If the scales are unbalanced, you know that the odd marble is in the group of 8 that you did weigh. This leads to two different cases.
Case 1
The scales balanced and the odd marble is in the group of 4 marbles that you didn’t weigh. Put 1 marble on each scale for your 2nd weighing. There will be 2 marbles that you do not weigh in the 2nd weighing. Either the scales will be balanced or unbalanced.
Case 1A:
The scales are unbalanced. Choose the marble from either the higher or lower side of the scale in the 2nd weighing, it doesn’t matter which one. Put it on the left side of the scale and a normal marble on the right side of the scale for the 3rd weighing. If the scales are unbalanced, the marble on the left side of the scale is the odd marble. If the scales are balanced, the marble you didn’t test is the odd marble.
Case 1B:
The scales are balanced on the second weighing so one of the two marbles you did not weigh is the odd marble. Put one of them on the left side of the scale and a normal marble on the right side of the scale for the third weighing. If the scales are unbalanced, the marble on the left side of the scale is the odd marble. If the scales are balanced, the marble that was not weighed is the odd marble.
Now for the tough part.
Case 2
4 marbles were placed on each side of the scale in the first weighing and the scales were unbalanced. Take 1 marble from each side of the scale and temporarily set them aside noting which marble was on the heavier side. Put all 6 remaining marbles on the same side of the scale keeping the 2 groups of 3 marbles separated and noting which group was the heavier in the first weighing. Weigh them against 6 normal marbles for the second weighing. Either the scales will be balanced or unbalanced.
Case 2A:
The scales are balanced in the second weighing. That means that one of the two marbles you did not weigh in the second weighing is the odd marble. For the third weighing, weigh one of them against a normal marble. If the scales are unbalanced that one is the odd marble. If the scales are balanced, the other marble is the odd marble.
Case 2B:
The scales are unbalanced in the second weighing. If the side of the scale with the 6 normal marbles went up, the odd marble is in the heavier set of 3, vice versa if the side with the 6 normal marbles went down. Choose 2 of the marbles at random from the group of 3 containing the odd marble and set the third marble aside. Put one marble on each side of the scale for the third weighing. If the scales are balanced, the marble that was set aside is the odd marble. If the scales are unbalanced and this group was lighter than normal in the second weighing, the marble on the side of the scale that went up in the third weighing is the odd marble. If this group was heavier than normal in the second weighing, the marble on the side of the scale that went down in the third weighing is the odd marble.
Phew! Having gone through all of that, there is one problem I know of. In Case 2, you need 6 normal marbles for the weighing but only 4 of the 12 were proven to be normal weight in the first weighing. I can’t get it to work out only weighing 4 marbles instead of 6. Oh well, that’s as close as I can get today.
1. Place 6 marbles on each side of the scale. Since only one marble is heavier, the heaviest side must contain the heavy marble.
2. Take the 6 marbles from the heaviest side from the first weighing. Place 3 on each side. Again, the side with the heaviest marble will be heavier.
3. Now you have 3 marbles, one of which is heavy, but only one use left of the scale. Pick any two marbles and weigh them. If one is heavier, that’s your marble. If they are equal, the third unweighed marble must be the heavy one.
The guy who solves the problem may be smart; the guy who posts it as a puzzle for other people to do is brilliant.