## Monday, June 20, 2011

### Fun Problems

A few days ago, Allison posted a problem you could really sink your teeth into and enjoy the puzzle of it all. It reminded me that it's been a while since I did this sort of brain challenge type problem just for the heck of it. Many years ago, I came across this problem:

Consider a square cake that's frosted on the top and sides. How do you cut it into 5 pieces so that everyone gets the same amount of cake AND the same amount of frosting? You must serve the whole cake. (be able to prove why it works) (added bonus: what if it's a rectangular cake?)

Now as a selfish person, I'm wondering if anyone has links to other cool such problems that make you think and tinker for a longer while than usual. Share! Share!

1. There are some fascinating problems at risps.co.uk, and these problems at Avery's blog are cool. Back in April I posted about this problem:

My favorite candy come in packages of 5 or 6. What's the largest number for which I cannot buy exactly that many candies? Can this be generalized?

And I loved this problem.

That should be enough to keep you out of trouble for a while. ;^)

2. Oh! Thank You! These look uber fun. Now I'm going away to churn on the "petal" problem .....

3. "Square cake" is not informative enough, as it does not give the height. For height 0, the problem is trivial (just 5 rectangular slices of the top).

For a cube, there is another easy, but tricky solution. Each frosted face has 1/5th of the icing, so let's cut in diagonally from the 8 frosted edges, making a pyramid for the top face and truncated triangular wedges for the side faces. All you need to figure out is how tall to make the pyramid to have 1/5 the volume of the cube. As long as that is less than the height of the cube you're set. Since the volume of a pyramid is 1/3 base area times height, the depth of the top piece at the center needs to be 3/5 of the side length.

4. You might like Wu Riddles. Some of the easier puzzles I give to my students after tests.

5. Gasstationwithoutpumps: hmmmmm, I guess I could refine it as "cutting the cake from the top down in a "normal" fashion. I'd have to think about the precision of words.

Mr. H: Thanks for another fun resource of brain candy.