Thursday, June 3, 2010

Genie and Hitch

Suppose Genie walks her dog Hitch on a trail where Hitch can run around without a leash. Genie wants to make sure Hitch gets a certain number of miles of running per day, but she doesn't want to walk that far herself. How long does Genie have to walk to make sure Hitch gets enough running?

Solution...

Thursday, April 15, 2010

More Unintelligent Design

Both eukaryotes and bacteria must transcribe the genes in their DNA into RNA transcripts in order to use them. The process starts out much the same in both kinds of organisms. In bacteria, the process is logical and efficient. In eukaryotic organisms like plants, fungi and animals (us), the process has gone hideously awry, and evolution has again employed a sloppy and inefficient fix.

Solution...

Wednesday, April 14, 2010

Unintelligent Design

I'm taking some biology classes. From a engineer's point of view, living things are not intelligently designed. Vestigial organs, vestigial genes, and a multitude of other obvious design flaws litter the biological landscape. For instance, we havean ugly hack embedded in our DNA replication.

Solution...

Tuesday, May 6, 2008

In Stitches

After doing the back-stitching on a few of the letters in this counted cross stitch pattern, I wondered if there would always be an even amount of stitches in the outline of a block shape.

Problem:

Define a "block shape" as any shape that can be created by coloring in squares on graph paper. Each square in a multi-square block shape must share at least one side with another square in that shape. Call the length of a square on the graph paper one unit. Prove or disprove: the perimeter of a block shape will always be an even number of units.

  • Base case: consider a one-square block shape. Its perimeter is 4 units - one for each side. Since 4 is even, the proposition holds for the base case.

  • Inductive hypothesis: assume that a block shape consisting of n squares has an even perimeter.

  • Deductive step: What happens when I add a square to the n-block shape?

    • If the new square is touching exactly one other square, then I am taking away 1 unit from the perimeter and adding 3 units. That's a net gain of 2 units, so the perimeter remains even.

    • If the new square is touching exactly two other squares, I am taking away 2 units and adding 2 units to the perimeter. The net change is 0, so the perimeter remains even.

    • If the new square is touching exactly three other squares, I am reducing the perimeter by 3 units and adding 1. The net loss is 2 units, so the perimeter remains even.

    • If the new square is touching exactly four other squares, I am taking away 4 units from the perimeter and adding none. The perimeter remains even.

    In each possible case, the perimeter of the (n+1)-block shape remains even.
Therefore, by the Principle of Mathematical Induction, the perimeter of a block shape will always be even.

Solution...

Tuesday, April 29, 2008

S.A.Tricky

Here's an S.A.T. problem that stumped my students and me (for a little while).

Problem:

a - b = 10
a2 - b2 = 50

Find b.



Edit: The original post contained a mistake. (10 + b)2 = 100 + 20b + b2, not 100 + 2b + b2.

Solution...

Thursday, April 24, 2008

IDEAs

Here's another S.A.T. problem.

Problem:

In the correctly worked addition problem to the left, A, B, D, E, I, and S each represent a different digit. What is the smallest possible value of D?

Solution...

Thursday, April 10, 2008

Prime Factorization

This problem comes form Project Euler. I solved it with a computer program written in C++.

Problem:

What is the largest prime factor of the number 600851475143?

Solution...