The following is a little preview a wonderful new book about
problem-solving. (See below for more information.)
This problem illustrates James Tanton's Problem Solving Tip #6:Eliminate Wrong Answer Choices
Solution: Let's figure out the answer by eliminating the ones we know are wrong. Choice (A) can't be right because then both children would be wrong. Choice (B) can't be right because Juan would be wrong, and we can eliminate choice (C) because it would mean that Lashana were wrong. But if we have eliminated (A), (B), and (C), then all we have left is (D). How could it be right? Choice (D) could be right if there were 3 horses on one side and 4 on the other, so that MUST be true!
Now use what you have learned to solve this harder problem:
The problems and the techniques are from the book Avoid Hard Work! … And Other Encouraging Mathematical Problem-Solving Tipsfor the Young, the Very Young, and the Young at Heart by Maria Droujkova, James Tanton, and Yelena McManaman. For more information, see http://naturalmath.com/avoid-hard-work/.
There are different ways to fill in the missing numbers on the trees, but importantly one thing is always the same. The numbers you end up with on the bottoms of the branches are always the prime factors of the number at the top. Think about the 40 tree for example. We can make a 40 tree in many different ways:
Notice that any way we construct the branches, we end up with 5, 2, 2, 2 at the bottom. The fact that we always end up with the same prime numbers, no matter how we factor a number is so important that it is called The Fundamental Theorem of Arithmetic.