logic puzzle

  • setting a question (for an exam)
    • There are two types of islands: Tribe A, which always says the right thing, and Tribe B, which always lies.
    • There are three natives; when I asked the first one his race, he answered in the local language, so I didn’t know.
    • I asked the second, “What did he say?” I replied, “He said he was Tribe B.” The third person countered, “No, no, he said he was Tribe A.”
    • Who is lying, the second or the third?
  • answer
    • The excellent part of this puzzle is asking who is lying, the second or the third person.
    • It is not possible to determine if the first person is lying.
    • If the first person is Tribe A, they honestly say they are Tribe A. If they are Tribe B, they lie and say they are Tribe A.
    • So there is no way to answer, “It’s Tribe B.”
    • From this we know that the second person is lying.

Is an irrational power of an irrational number ever a rational number?

  • Consider the numbers [$ x = \sqrt{2}^{\sqrt{2}}, y=x^{\sqrt{2}}=(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^2 = 2
  • Drawing useful conclusions without specifying whether x is a rational or irrational number
  • If x is a rational number, this is an example of “an irrational number whose irrational power is rational”
  • If x is an irrational number, then y is an example of “an irrational number whose irrational power is a rational number”
  • Therefore, we can say “there are rational numbers for which the irrational power of an irrational number is a rational number”.

Blind spot card 1027 #blind spot card with no picture yet


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