Balance Puzzles

For a worksheet with solutions, click here.

Factor Trees

Solution:  

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.

Tetrahedron Coloring Problem

For the solution, click here.

Honeybee Family Tree

For the solution, click here.  Thank you to Arushi G. for your help with this solution!

Invisible Square Problem

For the solution, click here.

Mobius Highway

Solution:  The blue car and red car will crash into each other.  They are in the same lane going in opposite directions.  The pink car is safely in the other lane.

One way to see this is to cut the two lanes apart.  You end up with a single strip of paper, but this time it is twisted twice, so it is no longer a Mobius strip.  (It has 2 sides rather than 1.)  You can see from the photo that the red and blue cars are on one side of the strip, heading toward each other.