A number theory problem by Arpit MIshra

Number Theory Level 2

Four servants of the King Octopus were conversing on a particular celebration night.

They said the following things: The Orange one said, “In total, we have 28 legs.” The Pink one said, “In total, we have 28 legs.” The Blue one said, “In total, we have 28 legs.” The Green one said, “In total, we have 28 legs.”

Now there is one thing particular about the servants. All of the servants have either 6 or 7 or 8 legs. Also, the ones with seven legs always lie while the ones with 6 or 8 legs are always truthful.

By reading the conversation, and with the given facts, can you find out the color of the servant who is telling the truth?

Green Pink and Blue Green and Orange Pink Blue All None Orange

Arpit MIshra by Arpit MIshra , 21, India

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2 solutions

Caleb Townsend
Apr 3, 2015

All four servants say the very same thing: therefore, they must all be telling the truth or all lying.

Assume they are telling a lie. They must all have 7 7 legs each, then, adding to 28 28 legs total. This contradicts the assumption that they were lying. All that remains is to prove that it is possible they are telling the truth.

Assume they are telling the truth. Then they must each have either 6 6 or 8 8 legs. Indeed, the combination 6 + 6 + 8 + 8 = 28 6 + 6 + 8 + 8 = 28 works. Since they cannot lie as proven above, this is enough to conclude they are all telling the truth, and two of them have 6 6 legs while the other two have 8. 8.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Arpit MIshra
Apr 2, 2015

The Pink servant is the truthful one.

In order to prove the same, let us take all of them one by one assuming that the selected one is telling the truth. This is because all of them are disagreeing with each other which means that three servants are lying while one is telling the truth.

Assume that the Orange servant is the truthful one. Under this condition, the orange one is having either 6 or 8 legs. And each of the other octopus is lying will then have 7 legs. Under this condition, the total number of legs will be 6 + 7 + 7 + 7 = 27 legs or 8 + 7 + 7 + 7 = 29 legs. But the Orange servant said that they have 28 legs all together. Thus he is lying.

Now if we simply follow the same logic with all other servants as well, we will come up to the conclusion that only the Pink servant is truthful.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

The Pink one said they have 28 28 legs total. If he is telling the truth, then so must everyone else be.

Caleb Townsend - 6 years, 2 months ago

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