From Alec Stein, inspired by conversations with Jesse Zymet and Adam Greenberg, comes a puzzle that is sure to make you lose your marbles:
At the Riddler Marble Shop, there are four enormous bags of marbles for you to acquire. They are labeled “red,” “green,” “blue” and “assorted.” Being the purist that you are, you want to select two bags of marbles that are not assorted, and you’d settle for some combination of red, green or blue.
However, noticing your interest in the bags, the shopkeeper alerts you. “Buyer beware,” she warns. “Some jerk switched around the labels on all four bags. Right now, every single bag is incorrectly labeled.” To give you a chance of properly identifying the bags you would like, she has kindly allowed you to take two — and only two — marbles out of any of the bags, one at a time.
How can you guarantee that neither of the two bags you take is assorted?
The shopkeeper said, “... every single bag is incorrectly labeled.” This yields nine possibilities.
| Labeled | |||||
|---|---|---|---|---|---|
| red | green | blue | assorted | ||
| Actual | 1 | assorted | blue | green | red |
| 2 | blue | assorted | green | red | |
| 3 | green | blue | assorted | red | |
| 4 | assorted | blue | red | green | |
| 5 | blue | assorted | red | green | |
| 6 | blue | red | assorted | green | |
| 4 | assorted | green | red | blue | |
| 5 | green | assorted | red | blue | |
| 6 | green | red | assorted | blue | |
Start with the bag labeled 'assorted'. WLOG, assume you get a red marble. This is the red bag.
Choose the bag labeled 'red'. There are three scenarios.
The bag labeled 'red' could be either the assorted or green bag. In either case, the bag labeled 'green' is the actual blue bag.
The bag labeled 'red' could be either the assorted or blue bag. In either case, the bag labeled 'blue' is the actual green bag.
The bag labeled 'red' must be assorted. This means the bag labeled 'blue' is the actual green bag, and the bag labeled 'green is the actual blue bag.
The following tree describes your decisions. Actual marble colors are in parenthesis.
There are times when the bag labeled a particular color is actually the assorted, and the bag labeled 'assorted is actually the bag of that particular color. In those three scenarios, you know all four bags, and thus can choose any three of the non-assorted.