Brain Teasers and Puzzles

Brain Teaser and Puzzles

When we were kids, Vake would challenge Patty, Sandy and me with brain-teasers and puzzles. Some were purely physical:  Can you put your feet behind your neck? Can you dance like a Cossack, starting in a deep squat, then bouncing up and down while you alternatively kick out your legs? Can you jump over a stick that you are holding in your hands in front of you? Patty did them all, easily. I got around to it.   Vake said Uncle John could do them—it’s no wonder that he needed hip replacement in his older years.

Then Vake presented mental problems.  “What’s the difference between walking and running?” We ran through the living room until we figured it out.  And, “You have a fox, a goose and a bushel of wheat that you need to carry across the river, but your boat will hold only two of them at a time. How do you get them across the river?  If you take the wheat, the fox will eat the goose.  If you take the fox, the goose will eat the wheat.  Hint:  The fox wouldn’t eat the wheat.

Brook and I worked ridiculously long on a puzzle over the holidays.  It started with my pointing out a quilt pattern called “One Thousand Pyramids.” I showed him a quilt top covered with regular triangles (all three sides the same side and each interior angle equaling 60 degrees) and asked, “Would there really be a thousand on one quilt top?”

We simplified the design to one of right triangles—one 90 degree and two 45 degree angles, with two sides the same size and one longer hypotenuse. That can be diagrammed as a square with in X in it. In a simple square with an X as diagonals drawn in it, you can find 8 triangles. If you make a grid of four, or nine, or sixteen squares, how many triangles can you find? 

You can diagram the squares and triangles and count them up, but the diagram gets more and more confusing each time you add a row of squares. Did I count that triangle yet, or not?

As it happens, there is high-powered math that gives you an answer to how many triangles can be found in the design, but probably only Brook and Dean would understand it. As it turns out, the problem is a classical math design called the “Tin Ceiling Puzzle.” I assume that it is named after the patterns stamped into the tin ceilings of old soda parlors and other such structures that you can find in the Midwest. Unfortunately, the math professor who posted a solution on the internet erred; one of his students had to straighten him out.

Brook explains (?):

Here is the solution to the version with squares:

n = the number of squares. Each square has lines from corner to to corner. Thus, eight triangles can be found in a single square. As you add squares to make a larger square, the number of triangles increase by more than just the number of triangles contained in the added squares:

for an even number of squares (n is even) the number of triangles is: 3n^3+9/2*n^2+n

for an odd number of squares (n is odd) the number of triangles is: 3n^3+9/2*n^2+n-1/2

I will tell you the difference between walking and running if you ask me. But I recommend that you run back and forth in your living room until you figure it out.