Tuesday, April 19, 2016

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.




Tuesday, March 1, 2016

Honeybee Family Tree

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

Monday, January 11, 2016

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.






Tuesday, November 24, 2015

Topologist's Map of the United States




To a topologist, two shapes are "the same" if one can be stretched, twisted, and distorted into the other shape without breaking or tearing it.  Shape, angle, size, distance, and position can be completely different, and yet the two shapes are topologically equivalent.

This is a topologist's map of the 48 contiguous United States.  While the states are not recognizable, you can work out which is which by how the states are connected to each other.  For example, which is the only state that borders exactly one state?  Can you find it on the topologist's map?  Can you identify all the states?  (One note:  Lake Michigan is included as one of the polygons, and the state of Michigan is represented as a single polygon, which was necessary to avoid some strange results.  So all together there are 49 polygons here.)

For a printable version click here.

For the solution, click here.