## Math Kangaroo’s Logic Puzzle

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

Share: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

## Leo:

Were any of the children transgendered?

22 May 2014, 2:02 pm## 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.

23 May 2014, 2:34 pm## 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”.

5 June 2014, 2:18 pm