## Tuesday, March 27, 2012

### One Coin Only

There are three clay containers. One of them has only pennies from the year 2005 in it, one has only pennies from the year 1975 in it, and the third has an equal number of each. They are labeled "2005," "1975" and "mixed," however, you know that the labels have been switched, so that each container is marked incorrectly. Can you properly label the containers without looking in them, and only pulling one coin out from one container?

1. Pick a coin from the container labeled "mixed" and examine it. Since we know that the containers are all labeled incorrectly, we also know that this is not in fact a mixed container--and thus that it contains coins from one year only: All of the pennies in the container labeled "mixed" are from the same year as the penny you pull.

Suppose you pull a penny from year A from the container labeled "mixed", and determined then that the "mixed" container contains all pennies from year A. Consider the container labeled with year B. This container cannot contain all pennies from year B, since we know that it is mislabeled; and it cannot contain all pennies from year A, since we've already found those pennies; so we conclude that this is in fact the container containing coins from both years. The pennies from year B are to be found in the container labeled with year "A".

Results/Implementations:

-Pull a 2005 penny from "mixed"-
"mixed": 2005 pennies only
"1975": both 2005 and 1975 pennies
"2005": 1975 pennies only

-Pull a 1975 penny from "mixed"-
"mixed": 1975 pennies only
"1975": 2005 pennies only
"2005": both 2005 and 1975 pennies

2. As always, oudeis, you impress me with the depth of your answer!

3. Aww darn! For once I was like "Oh yeah! I know this one!!!" Only to see I wasn't the first. Although, I'm pretty sure oudeis' answer is probably better than mine would've been :)

4. Shucks... thanks guys :)

5. Pull a coin from the mixed jar. You know all jars are labeled correctly so whatever year comes out of mixed goes there. Consequently, the year that doesn't belong to the jar originally labelled "mixed" belongs to the other unidentified jar. Thus, leaving the last jar as mixed.

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