Problem Solving and Computing (CSC-103 98S)

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# Problem Set 3: Mostly Geometry Problems

Assigned: Tuesday, February 3, 1998
Due: Tuesday, February 10, 1998

Do problems 1, 4, 6, and 7

1. Triangles on the chessboard

[Original problem from the notes of Emily Moore. Modified by Sam Rebelsky.]

A chess board is an array of 8 by 8 squares. In an earlier exercise, we reflected on the number of squares one could draw on the chess board. In this exercise, we will consider the number of triangles we can draw on the chess board.

For the purposes of this problem, the chessboard is arranged so that the squares are aligned with the horizontal and vertical axes (i.e., the board is "straight").

How many unique triangles can we draw on the chessboard if each corner of every triangle must fall at the corner of a square on the chessboard?

How many unique triangles can we draw on the chessboard if each corner of every triangle must fall at the corner of a square on the chessboard, one edge of each triangle must be horizontal, and one edge of each triangle must be vertical?

How many unique triangles can we draw on the chessboard if each corner of every triangle must fall at the corner of a square on the chessboard, one edge of each triangle must be horizontal, one edge of each triangle must be vertical, and the horizontal and vertical edges of each triangle must be of equal length?

2. Three generations

[From the notes of Emily Moore.]

When I am as old as my father is now, I shall be five times as old as my son is now. By then my son will be eight years older than I am now. The combined ages of my father and myself are 100 years. How old is my son?

3. Division by 9

It is a "well known" fact that a number is divisible by nine only when the sum of the digits in the number is divisible by nine. Is it true? Can you figure out why? Is it true for any other number?

4. Counting faces

[From the notes of Emily Moore.]

The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of these vertices, four faces meet, and at each of the other three vertices, six faces meet. How many faces does the figure have?

5. A surprising sphere

[Based on the notes of Emily Moore.]

The surface area and volume of a certain sphere are both four-digit integers times pi. What is the radius of the sphere?

6. Changing the base

[From the notes of Emily Moore.]

An isosceles triangle has a 10-inch base and two 13-inch sides. What other value can the base have and still yield a triangle with the same area?

7. Indirect directions

[Based on a question posted by Scott Imig when reflecting on infinity.]

Scientists have recently discovered a previously unknown animal wholse behavior is closely tied to the Earth's magnetic field. One surprising characteristic of this animal is that it cannot walk directly East. In fact, if it wishes to head East, it can only walk along paths that are 60 degrees North of East or 60 degrees South of East., as in the following figure:

```  legal
/
/
/ 60
--------------> East (illegal
\ 60
\
\
legal
```

In order to get one mile East, the animal can walk one mile "Northeast" and one mile "Southeast". It then takes the animal two miles to go East by one mile. The animal might try to improve this path by turning every half mile, rather than every mile (i.e., 1/2 mile "Northeast", 1/2 mile "Southeast", 1/2 mile "Northeast", 1/2 mile "Southeast"). What distance does the animal walk in this case?

The animal might continue improving its path, turning every 1/4 mile, every 1/8 mile, and so on and forth. Can you come up with a formula for the distance it travels in each case?

If you've dealt with limits, consider the following: if we consider the limiting case (turning in increasingly minute fractions of a mile), what distance does the animal go when trying to move one mile East? Why might someone suggest that this is counter-intuitive?

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