The Race

Harel has challenged Lisa to a race. He plans on riding his bicycle, but Lisa will be driving a car. Both contestants will be traveling at a constant speed throughout the race, but Lisa's car is significantly faster than Harel's bicycle. Say that Lisa's car is k times as fast as Harel's bicycle, but k>1. This challenge seems to be not too bright of a move on Harel's part, but he has added the condition that he be given a 5-minute head start. Assume that the finish line is very, very VERY far away -- say a zillion miles.

1. Who do you intuitively think will win the race?

Well, just as Lisa was smugly imagining her victory, Harel burst her bubble:

"Wipe that silly grin off your face, Lisa. There is no way you can win this race; I've tricked you. consider the diagram below of our positions on the track. H0 represents my position at the starting line. After t1=5 minutes, my position will be H1, but you will still be at the starting line, position L1=H0.

At sometime, call it t2, you will reach the position L2=H1, but by then I will be farther ahead, at position H2. Then by the time you reach position L3=H2 (call this time t3), I will be even farther ahead, at position H3. This process will continue indefinitely, so that at each time tn, you will be at Ln=Hn-1, but I will be at Hn which is farther ahead. Therefore, you can never catch up with me, no matter how fast your car is. Victory will be mine!"

Now Lisa has gone into seclusion; Harel's argument seems entirely logical, yet it contradicts her firm belief that fiery red hot-rods should beat bicycles in races. She feels forced to choose between mathematical logic and her own concept of the real world. you can help to save her sanity if you find a flaw in Harel's argument.

©Copyleft 1995 by Lisa Orlandi. "Share and Enjoy." Inspired by one of Zeno's Paradoxes.
Note: Lisa Orlandi and Harel Barzilai were the instructors of the first two sections of projects-based Math 112 the Math Department offered, back in the Spring of 95. This Activity was modified from one of Zeno's Paradoxes, by Lisa Orlandi.