Tuesday, October 09, 2007

October 2007: On the right track (solution)

The right-hand wheels travel 2π" farther than the left-hand wheels.

The two pairs of wheels travel the same distance along the straight tracks, so we may as well ignore them. We can take the curved tracks and reassemble them to form a circle. The circumference of the inside rail is 2πr and the circumference of the outside rail is 2π(r+1), so the difference is 2π". (If the track is not convex, then each right-hand curve cancels a left-hand curve, so we can toss them both out, leaving a single circle.)

Math fact: This applies much more generally. If a vehicle drives counterclockwise along any curvy route so that it loops back to the starting position without ever crossing its tracks (think of a Grand Prix race track), then the right wheels travel 2πb farther than the left wheels (where b is the distance between the wheels).

Correct solutions: Ritoban Basu Thakur

Wednesday, October 03, 2007

October 2007: On the right track

Your young cousin has made an elaborate train track for his toy trains. He used track pieces of two varieties--ones that are straight and ones that are arcs of a circle. The gauge of the track (the distance between the rails) is 1". Suppose he drives his train around the track one time counterclockwise. How much farther did the right wheels travel than the left wheels? (You may assume that the track is convex, but in fact, you don't need this assumption--you just need there to be no crossovers.)