Math Kangaroo’s Logic Puzzle

My AMSA students loved the following puzzle from the 2003 Math Kangaroo contest for grades 7-8:

The children A, B, C and D made the following assertions.

  • A: B, C and D are girls.
  • B: A, C and D are boys.
  • C: A and B are lying.
  • D: A, B and C are telling the truth.

How many of the children were telling the truth?
A) 0   B) 1   C) 2   D) 3   E) Impossible to determine



  1. Leo:

    Were any of the children transgendered?

  2. anon:

    D is clearly lying. One or both of A and B are lying. If both A and B are lying, then C is telling the truth. If one of A and B is lying, then C is lying. Either way one person is telling the truth.

  3. Douglas J. Keenan:

    The sought-for answer is, presumably, 1, but that is incorrect. As worded, either 1 or 0 people could be telling the truth; hence the correct answer is “Impossible to determine”.

    The difficulty here is that someone might not be lying but could still be wrong. For example, both A and B could be wrong, but honestly mistaken; thus neither is lying, and so C is wrong too. In this example, then, there are 0 people telling the truth.

    Perhaps the third assertion should be reworded, e.g. “Neither A nor B is telling the truth”.