Recently I asked a curious question about this curious set of labelled jars. Assuming there is a jar of red meeples, a jar of green meeples and a jar of mixed red and green meeples, and additionally that a friend of mine swaps the contents of the jars around so that none of the labels match anymore:** What is the minimum number of meeples I have to look at to be certain that I know which jar is which?**

As a couple of commenters pointed out, and as I think most people would deduce, **the answer is precisely 1**. All I have to do to be certain of the contents of all the jars is to take a single meeple from the jar labelled as mixed. That jar must contain, since all the labels are wrong, either all red or all green meeples. A single meeple plucked from its interior will tell me which, and I can then immediately fix the contents of the other jars using the key knowledge that all the labels are wrong. If I pluck a red meeple from the mixed-labelled jar then I know not only where all the red meeples are, but also that the mixed meeples must be in the green-labelled jar, and hence that the green meeples must be in the red-labelled jar. There is no other possible configuration. (If the red meeples are in the mixed-labelled jar and the mixed meeples were in the red-labelled jar, that that would mean that the green meeples were still in the green-labelled jar, and we know that’s not possible!)

The idea for the puzzle was, like my first Puzzling Meeple puzzle, borrowed from Tanya Khovanova’s excellent blog. However, I did mix things up a bit by asking a second question, in which there were now four jars — containing red, yellow, green and mixed meeples — the contents of which, once again, had been switched around so that none of the labels were correct. What then is the minimum number of meeples needed to be plucked?

It turns out, as commenter Nunya Bidness correctly stated, that the answer is indeterminate given my statement of the problem. Which it to say, that unless you know how many meeples are in the jars you can’t give a definite answer. However, if I were to tell you that I know that each jar contains exactly 100 meeples, and reiterate the constraint that the mixed-labelled jar does indeed contain at least one meeple of each colour, then the answer to the question “How many meeples must I look at?” is precisely 100! (At least it is if I allow the concession that you are able to feel around in the bottom of a near-empty jar and determine when only 1 meeple remains.)

However, a correct answer must state the minimum number I must look at to be certain of knowing — for all possible configurations, and all possible ways in which I might, at random, pull meeples from the jars. The question is perhaps more clearly stated like this: “What is the minimum number of meeples I must look at if I were to be supremely unlucky?”

Let’s look at my best strategy. As before, I start by pulling one meeple from the mixed-labelled jar. Since this jar must contain only red, yellow or green meeples this one meeple will tell me the colour of all the meeples this jar contains. One down, three to go!

Now let’s look at the possible contents of the jars, and let’s assume for argument’s sake that I pulled a red meeple from the mixed-labelled jar (if I pulled a yellow or green meeple the specific configurations are different, but the logic is exactly the same). There are three configurations in which the red meeples are in the mixed-labelled jar and the other three jars all have the wrong labels.

We can observe that if I were, say, to now pluck a red meeple from the red-labelled jar then the configuration of the jars would be fixed: it would have to be configuration A. This means that if I’m lucky, the minimum number of meeples I look at is only 2, but that is definitely not the minimum number I must look at to be certain in all possible cases. Far from it!

For example, I could keep plucking meeples from the green-labelled jar and keeping finding yellow ones and I’d be none of the wiser: all three configurations allow for some yellow meeples to be in the green-labelled jar. Indeed, even if I continued to take meeples from the green-labelled jar until all but two remained in the jar it’s possible that all of them could be yellow and hence that I couldn’t determine anything about the configuration. It’s true that if I then took the last-but-one meeple out and it too was yellow that I’d be able to rule out configuration C (since I know that the mixed-meeple jar originally contained at least one meeple of each colour), but I’d be no closer to distinguishing A from B!

So, what is my best strategy, assuming the worst possible luck? It is to draw meeples from the red jar (that is, from the jar matching the colour of the single meeple I took from the mixed-labelled jar), and keep drawing meeples from this jar until I know this jar’s contents. And the worst-case scenario is that I may have to draw all but one of the meeples from this jar to do so!

As I say, I didn’t tell you how many meeples were in any of the jars, so the numerical answer to the original question is indeterminate. However, you can describe the procedure, which is to pull 1 meeple from the mixed-labelled jar, plus all-but-one of the meeples from the jar matching the colour of that first meeple. In the case where the jars all contain 100 meeples, the answer is therefore 1 + 99 = 100 meeples. Which means I think that, more generally, it would be proper to say that the numerical answer is, interestingly enough, “the average number of meeples in each jar” and that this answer is correct whether or not that average is known.

At least, I *think* that’s the answer! If you think different, do let me know!

Seth Jaffee said...

April 21, 2011 6:23 pm

Well, there was no restriction that the same number of meeples existed in each jar, nor that the same number of each color meeple existed at all (just that there's at least 1 of each color).

So the general solution would be something like "1 from the mixed labeled jar, and all but 1 from the jar of that color," wouldn't it?

Brett said...

April 21, 2011 8:35 pm

Seth: Everything you say is spot on, of course. I was just using the '100 meeples in every jar' thing as an example. I should point out that when I asked the question I hadn't actually worked out the answer ;-)

dagautier said...

January 27, 2013 9:06 pm

Where may I find this kind of "jar" ? thank you :)

55cards said...

January 28, 2013 10:43 am

dagautier: I'm afraid these jars are not available for sale! I added the meeples in Photoshop!

dagautier said...

February 05, 2013 10:15 am

@55cards:disqus argh :o/

Nicely done indeed :)

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