**Assigned:** Thursday, April 30, 1998

**Due:** Thursday, May 7, 1998

*As always, you should feel free to work on these problems in
any order.*

**1. The Tethered Goat, Revisited**

[From the notes of Emily Moore. Modified by Sam Rebelsky.]

A goat is tethered to a corner of a regular hexagonal house which is in the middle of a grassy plain. The hexagon measure ten feet on a side. The rope stretches half way around the hexagon. Find the area of grass that the goat can reach.

**2. Full-length Mirrors**

[From the notes of Emily Moore.]

What is the height of the shortest wall mirror in which you can see both your hair and your shoes at the same time?

**3. Faulty Bricks**

[From the notes of Emily Moore. Modified slightly by Sam Rebelsky.]

Many building bricks are twice as long as they are wide. I need to pile some up in a rectangular pile until I am ready to use them, and I would like to do it in such a way that each layer is free of any fault line going between two layers.

What sizes can I make the layers?

By *fault line* we mean a continuous straight line from left
edge to right edge that runs between two layers.

**4. Lunar Distance**

[From the notes of Emily Moore.]

A quarter (a 25 cent piece) is 3/4 inch in diameter, and when placed 7 feet from the eye will just block out the disc of the moon. If the diameter of the moon is 2160 miles, how far is the moon from the earth?

**5. Tall Trees**

[From the notes of Emily Moore.]

Devise a method for measuring the height of a very tall tree. (You are not allowed to climb the tree.) Explain your method carefully.

*If weather permits, we will use this method to measure trees around
campus next week.*

**6. Arithmagons**

[From the notes of Emily Moore. Modified by Sam Rebelsky.]

A secret positive integer is assigned to each vertex of a triangle. On each side of the triangle is written the sum of the secret numbers at its ends.

For example, we might have a triangle with sides labeled 11, 18, and 27. What are the secret numbers from which these numbers are derived?

Find a simple rule for revealing the secret numbers. What triples of numbers are possible for labels for the sides? What are impossible?

Does it make a difference that the integers are positive? Why or why not?

Generalize this question to other polygons. Try to find other questions you might ask.

**Disclaimer** Often, these pages were created "on the fly" with little, if any, proofreading. Any or all of the information on the pages may be incorrect. Please contact me if you notice errors.

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