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True, False, I Do Not Care...

Consider this list of twelve statements:

  • Precisely one of these statements is false.
  • Precisely two of these statements are false.
  • Precisely three of these statements are false.
  • Precisely four of these statements are false.
  • Precisely five of these statements are false.
  • Precisely six of these statements are false.
  • Precisely seven of these statements are false.
  • Precisely eight of these statements are false.
  • Precisely nine of these statements are false.
  • Precisely ten of these statements are false.
  • Precisely eleven of these statements are false.
  • All twelve of these statements are false.
Which statements are true? Explain.

Which statements are false? Explain.

Any statements that could be true or false? Explain.

 

Source: James Tanton's "A Dozen Questions About a Dozen," Math Horizons, April 2007, pp. 12-16.


Hint: Consider a smaller problem:

  • Precisely one of these statements is false.
  • Exactly two of these statements are false.
  • Does this help? Can you transfer your reasoning to the full set of twelve statements?

     


    Solution Commentary: First, since all of the statements contradict each other, it is impossible for two statements to be true.

    But, if all twelve statements are false, then the last statement must be correct, which leads to a contradiction.

    Thus, the only option is for eleven statements to be false and one statement to be true...so now, which one is the true statement?