Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem24

Levels 9&10 Video Solutions 2010 problem24

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Question number 24. An underwater kingdom is inhabited by creatures with either 6, 7, or 8 arms. Those with an odd number of arms always lie, while those with an even number of arms always tell the truth.  One day, four creatures met. The blue creature said, Together we have 28 arms. The green creature said, Together we have 27 arms. The yellow creature said, Together we have 26 arms. And the red creature said, Together we have 25 arms.  How many arms did the red creature have?  What we note here, first of all, is that each of the four statements claims a different number of arms. Therefore, not all of them can be correct at the same time.  And, in fact, only one of them can be correct if any of them are correct at all. So what we conclude is that at least three of the creatures are liars.  At least three creatures lie, and 21 arms, 3 times 7, belong to the liars. Now we can analyze each of the remaining statements in this context.  So the blue creature says that there are 28 arms. And if 21 of them belong to liars, then it must be that the last creature here has 7 arms.  And that tells us that there are four liars. So that tells us that the blue creature is lying. He is telling a lie, but the number comes out to 28, which is the correct sum.  So we conclude that blue lies, and one other creature tells the truth.  In other words, there are exactly three liars.  And we know that blue here lies. OK, so let's move on. Let's move on to the green creature. And green says that there are 27 arms.  And that has to be divided into 21 plus 6, which is OK. And it is possible to have 6 arms and be truthful. So we conclude that green is telling the truth.  And at this point, we can answer the question, if green is telling the truth, he is the only creature of the four that is telling the truth.  And therefore, the red creature must be a liar. And if he is a liar, he must have an odd number of arms, so 7.  But we can finish the analysis either way and go on to the yellow creature. So here in orange, I'll say the yellow creature claims there are 26 arms.  And that must be divided into 21 plus 5, which is not possible because there are no creatures with 5 arms. And finally, the red creature claims that there are 25 arms.  And again, that must be divided into 21 plus 4, which is not possible because there are no creatures with 4 arms. So both of these are liars.  And so now we know that yellow, red, and blue are liars. The green made a true statement. It is OK to have 6 arms and then still have the correct total. So make a true statement.  So we found out that, therefore, red is a liar and must have 7 arms, so the answer is B.

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