Thursday, February 11, 2010

Vancouver Olympics Puzzle

Weather has not collaborated with the Olympic schedule this year, and instead of Vancouver a few feet of snow have been dumped on Washington DC. With not too much pressed white powder to slide on, skiers felt strong friction on the slopes. One of the athletes reported that in the first second of her descend she moved just 10 feet, in the second only 5 feet, in the third 2.5 and so on slowing down by half every second. Will she reach the finish line, which is 100 feet down from the start, in less than a minute?

Submit your answer on our Family Puzzle Marathon Be first to solve three puzzles and get a prize!

Kim said...

Not going to make it. Not even in 10 minutes or 100 minutes... At the rate at which the poor skier is decelerating, they are going to start barely moving after a couple of more seconds, and they are never going to quite make it beyond the 20 foot mark.

Maria said...

Kim - you are brilliant but I can't count your answer. You just solved the hard puzzle. I mentioned in the newsletter that one person can submit answer for only one out of three posted puzzles. So, if anyone is interested, this puzzle is still open. Besides, can someone come up with a nice convincing explanation for this one?

W said...

Well, this doesn't seem fair now that Kim's already answered but I can prove mathematically that it won't go beyond the 20 foot mark.

Since we start out at 10 feet / sec and then decrease by 1/2 every sec, this is a geometric progression with a, the first number = 10 and r = 1/2 (the ratio between two consecutive numbers in a sequence).

The formula for finding the sum of the geometric sequence (infinite) is:

a / (1 - r)

so plugging in the numbers..

10 / (1 - 1/2) = 10 / (1 / 2) = 20

So even if she was given an infinite amount of time, she would only reach the 20 foot mark (and never get past that).

Another way of thinking about it is that she reaches the halfway mark between 0 and 20 feet in 1 sec. Then, she reaches the halfway mark between 10 and 20 feet in 1 sec. Again, she reaches halfway between 15 and 20 feet in 1 sec. Since she only travels half the distance between where she is now and 20 feet, she will only reach 20 feet after an infinite amount of seconds has passed.

W said...

I forgot to point out that finding the sum of the geometric infinite sequence only works if -1 < r < 1.

Phil said...

This is a classic limit problem. The skier can only ever get to twice the distance traveled in the first second, and only if given infinite time. Kim's answer is a _little_ bit off with the use of the word "beyond", when in fact the skier will never go beyond the 20ft mark.

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 ....

Maria said...

Beautiful explanation by W. who gets a 10th puzzle point and will be turned into a puzzle at the next suitable occasion.

Phil is also correct, but W. was first, not counting Kim. Very suspicious - this is not the first time W. and Phil submit their answers within three minutes interval.