## 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

Share:

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”.

4. #### Lori:

Wow, this question seems a bit confusing for a middle schooler. We’ve tried out Math Kangaroo before but it didn’t work out as much as we had hoped. Beestar has been a pretty good option with their online competitions.