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The puzzle is as follows:

In a certain kindergarten there are 23 children. Some of them wear a green t-shirt, others a yellow t-shirt and others blue t-shirt. A tv reporter who was on the site doing a documentary made them the following three questions:

Are you wearing a green t-shirt? 17 of them answered yes.

Are you wearing a yellow t-shirt? 12 of them answered yes.

Are you wearing a blue t-shirt? 8 of them answered yes.

It is known for a fact that in such peculiar kindergarten, those who wear a green t-shirt always tell true statements, while children wearing a yellow shirt alternate the truth value of their answers, and those who wear blue t-shirts always tell false statements.

With all of this information, how many of those children were wearing a yellow t-shirt when interviewed by the reporter?

The alternatives given in my book are as follows:

  1. 12
  2. 10
  3. 13
  4. 14

I'm confused exactly on how to approach this question methodically without having contradictions.

I'm assuming that from the given information, those who are wearing a green t-shirt will say true, but there are others who will also say this, so it is not possible to know from this clue alone who is wearing green.

The other clue mentions that those who are wearing yellow colors alternate their truth value. From this I'm assuming that we can be sure that at least 6 are wearing yellow color.

But there is more.

The other half could only be wearing Blue because if they were wearing green it meant that they are telling true and that must be their color.

Those who are wearing blue cannot be blue because that could cause a contradiction, so this means they are wearing yellow.

Therefore, this means:

8+6=14.

But is this the right way to approach this puzzle? Or am I wrong in my deduction?

I also noticed that with the given clues, it is not possible to know how many are blue. Am I right with this?

Can someone guide me in a better way to think this problem?

For reference, this riddle comes from my Reason and logic from 2000s which seems to have puzzles from a reprinted version of Martin Gardner's 1970s, Puzzle carnival.

So all and all, can someone help me? Is there a way to do this without just getting confused?.

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I think this answer is entirely solvable!

You are correct - there are 14 Yellow shirts, also 5 Green and 4 Blue.

How do we start?

Instead of splitting the children into 3 groups, split them into 4 groups. Green who always tell the truth, Blue who always lie, Yellow-T who start in truth-telling mode, and Yellow-L who *start in lie-telling mode.

This means we can analyse the 3 questions fully.

Question 1:

G will answer "Yes" because they tell the truth.
B will answer "Yes" because they are lying.
Y-T will answer "No" because they tell the truth this time.
Y-L will answer "Yes" because they lie this time.

Meaning your total "No"s are only in Y-T which is 6.

Question 2:

G will answer "No" because they tell the truth.
B will answer "Yes" because they are lying.
Y-T will answer "No" because they now have to lie, because they told the truth last time.
Y-L will answer "Yes" because they now tell the truth.
Meaning your total "No"s are 11 and are either Y-T and G, but we know Y-T is 6, meaning G is 5.

Question 3:

G will answer "No" because they tell the truth.
B will answer "No" because they are lying.
Y-T will answer "No" because they tell the truth again!
Y-L will answer "Yes" because they are back to lying.

Meaning your total "Yes"s are only in Y-L which is 8.

Therefore:

You know all of the totals. 6 + 8 = 14

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  • $\begingroup$ I think you have a typo: 8 people answered "Yes" to the third question. $\endgroup$
    – user3294068
    Jan 13, 2021 at 19:15
  • $\begingroup$ Yes - Thank you for that! $\endgroup$
    – Graylocke
    Jan 13, 2021 at 22:22
  • $\begingroup$ @Graylocke I'm getting the idea that the three questions are made at the same time to all children thus this makes it possible to know which colors go to which. It isn't exactly the sort of diagram which I was intending to ask but it is good and logical so I'm accepting your answer. Thanks for that. $\endgroup$
    – Chris Steinbeck Bell
    Jan 20, 2021 at 8:42
  • $\begingroup$ No worries. Though I think you may be incorrect in your assumption. For one thing, I don't really know how you would ask 23 children 3 questions simultaneously (and expect answers you can understand!) For a second thing,the information that yellow-shirts alternate in truth-telling would be useless information to put into the problem otherwise. I do agree that the question could be worded more clearly, however. $\endgroup$
    – Graylocke
    Jan 20, 2021 at 9:29
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You asked for methodical:

First let's get one thing straight: Each kid answers each of the three questions. And. Each yellow kid can be in two states at the beginning: either lying then it will tell the truth to the second question and lie again at the third or it can be truthful first then it will lie to the second question and be truthful again for the third. Let's call these to groups ltl and tlt.

As you have correctly established the kids that say yes to blue must all be yellow and must be ltl. In fact these kids will have answered with yes to all three questions. Similarly, the remaining 4 that said yes to yellow must be the blues, because they cannot be tlt nor green and ltl is already accounted for. Finally, both groups already established will have lied yes to question one leaving 5 more to say yes to green. These cannot be tlt hence must be the greens. We have now assigned 17 of the 23 leaving 6 that can only be tlt.

The total for yellow is therefore:

ltl + tlt = 8 + 6 = 14

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