***UPDATE*** I have collaborated with the talented TED-ED team to bring you a fully animated version of this riddle. Continue reading for the video!
I learned an interesting riddle a few years ago. It goes like this:
“You have 12 marbles and a balance scale. One of the 12 marbles is inconsistent with the others, meaning it could be heavier or lighter than its peers of normal weight. You are allowed to use the balance scale exactly 3 times to identify which of the 12 marbles is irregular AND determine whether it is heavier or lighter than normal.”
I also learned a solution awhile back, but had a difficult time internalizing it. The solution was involved and in my opinion, not very intuitive. So I forgot about the problem, until recently, when my dad came up with a creative approach to solving it that’s both accurate and easy to understand.
You can see the solution in action in the videos below!
Collaboration with TED-ED
My original video
As promised in the video, I also want to provide a written solution.
The hardest part about this problem is keeping track of the various scenarios. To help us stay organized, we can create a key as follows:
Green = Normal weight
Yellow = Lighter if known
Blue = Heavier if known
White = No information
Note that in the TED-ED video, “0”, “+”, and “-” are used in place of the green, blue, and yellow, respectively, but the principles outlined below still apply.
Setup: We start with 12 white marbles, because without weighing any of them, we have no information regarding their relative weight.
- Divide the 12 marbles into three groups of four
- Weigh two of the groups
- Now, we break down the possible outcomes
Outcome 1: Both sides weigh the same
- Color the eight marbles on the scale green, because they are all of normal weight
- Take three of the remaining four white marbles and weigh them against three of the eight green ones.
Sub-scenario 1: The two sides weigh the same. We now know that the three white marbles are normal, so we can color them green. We also know the last marble is the irregular one. To determine whether it’s lighter or heavier than normal, weigh it against a green marble.
Sub-scenario 2: The two sides don’t weigh the same. Based on this result, we know whether our irregular marble is lighter or heavier than normal. For example, if the three white marbles are lighter than the three green ones they’re weighed against, we can color the white marbles yellow. To identify the culprit, weigh two of the yellow marbles against each other. The lighter one is the oddball. If the two weigh the same, then the third, non-selected yellow marble is irregular. The same principle applies with the blue marbles, except in the final weighing, the heavier of the two is the oddball.
Outcome 2: After the initial weighing, the two sides don’t balance
- Color the four unweighed marbles green, because they are of normal weight
- On your scale, there’s now a heavier and a lighter side. Color the heavier side blue and the lighter side yellow
- Setup the second weighing. In this example, we will weigh three yellow and one blue marble against three green and one yellow.
Sub-scenario 1: The side with the three yellow and one blue marble is heavier.
This means the oddball is either the blue marble on the left side, or the yellow on the right. To determine the culprit, weigh one of them against a green. If the two balance, then we know the remaining unweighed marble is the oddball
Sub-scenario 2: The side with the three yellow and one blue marble is lighter.
This means the oddball is one of the three yellow marbles. Weigh two of them against each other. If they’re not balanced, the lighter of the two is the culprit. If they are balanced, then the third unweighed yellow marble is irregular
Sub-scenario 3: The two sides weigh the same.
This means the oddball is one of the remaining three blue marbles that we did not include in our second weighing. We follow the same procedures outlined in sub-scenario 2 of outcome 2 to uncover the irregular marble.
There you have it – the full solution to this popular riddle broken down by scenario.
Do you know of any other interesting brain teasers? Please share them in the comments below!