Answer to Problem of the Week for 10/13/03

Answer to October 13, 2003 Problem by Bob Stanton

Winchester Transit Problem The Winchester Transit Authority operates a bus line with seven stops -- A, B, C, D, E, F, and G. Here are some clues about the seven bus stops: From A to B is 2 miles. From A to D is 7 miles. From C to E is 3 miles. From C to G is 7 miles. From E to F is 2 miles. From E to G is 4 miles. B is 4 miles due southwest of C. D is 5 miles due southeast of B. F is 4 miles due northeast of D. The bus begins its route at A and passes every stop once before returning to A, following the shortest path possible. Unlike most buses, this one always travels in a straight line between stops, and straight roads exist between every possible pair of bus stops. If the bus stops at B sometime before it stops at C, in what sequence does the bus pass the stops?
Solution to the Problem: The bus route is A-B-D-G-F-E-C and then back to A. Use a piece of graph paper to help solve this problem. Use the last three clues to begin plotting the points C, F, D, and B. From these last three clues, CFDB must be a rectangle. Hence, CF is 5 miles long and parallel to BD. Since CE is 3 miles and EF is 2 miles, E must be a point on line CF. CG is 7 miles, CE is 3 miles, and EG is 4 miles, so CEFG is a straight line. G must be 2 miles southeast of corner F of the rectangle. AD is 7 miles, AB is 2 miles, and BD is 5 miles, so ABD is a straight line, and A is 2 miles northwest of corner B of the rectangle. It's now easy to see that the shortest route that passes every bus stop travels from A through B, D, G, F, E, and C, or the reverse. Since the bus stops at B before C, this is in fact the route.

Winchester Transit Problem

The Winchester Transit Authority operates a bus line with seven stops -- A, B, C, D, E, F, and G.
Here are some clues about the seven bus stops:

From A to B is 2 miles.
From A to D is 7 miles.
From C to E is 3 miles.
From C to G is 7 miles.
From E to F is 2 miles.
From E to G is 4 miles.
B is 4 miles due southwest of C.
D is 5 miles due southeast of B.
F is 4 miles due northeast of D.

The bus begins its route at A and passes every stop once before returning to A, following the shortest path possible.
Unlike most buses, this one always travels in a straight line between stops, and straight roads exist between every possible pair of bus stops.

If the bus stops at B sometime before it stops at C, in what sequence does the bus pass the stops?

Solution to the Problem:

The bus route is A-B-D-G-F-E-C and then back to A.

Use a piece of graph paper to help solve this problem. Use the last three clues to begin plotting the points C, F, D, and B. From these last three clues, CFDB must be a rectangle. Hence, CF is 5 miles long and parallel to BD. Since CE is 3 miles and EF is 2 miles, E must be a point on line CF. CG is 7 miles, CE is 3 miles, and EG is 4 miles, so CEFG is a straight line. G must be 2 miles southeast of corner F of the rectangle.

AD is 7 miles, AB is 2 miles, and BD is 5 miles, so ABD is a straight line, and A is 2 miles northwest of corner B of the rectangle.

It's now easy to see that the shortest route that passes every bus stop travels from A through B, D, G, F, E, and C, or the reverse. Since the bus stops at B before C, this is in fact the route.

Correctly solved by:

1. Akash Patel	Columbus, Georgia
2. Jeffrey Gaither ***	Winchester, Virginia
3. Walt Arrison ***	Philadelphia, Pennsylvania
4. Richard Johnson	LaJolla, California
5. Keith Mealy	Cincinnati, Ohio
6. John Funk	Ventura, California
7. James Alarie ***	University of Michigan -- Flint, Flint, Michigan
8. Jim Kennedy	Limerick, Ireland
9. G. Beavers	----------
10. Chris McCormick	Winchester, Virginia
11. Andrew Oliver	Winchester, Virginia
12. Viktor Bergqvist	Tullängen, Sweden

*** sent in the correct distance for the trip of 22.944 miles

Answer to October 13, 2003 Problem by Bob Stanton

Correctly solved by:

Answer to October 13, 2003 Problem
by Bob Stanton