Kids attached a few sleds together and were pulling each other on the fresh snow from one end of the street to another. A discussion started about how long is their sled train. Our great puzzle solver - W - passed by. He told the kids that his step is two feet long and bet that he can measure the sled train by walking along it. However, as soon as he started walking along the train, a mischievous kid started pulling the train in the direction of his walk. W decided to ignore the distraction and continued walking with the same speed while the sled train continued to move. When he finally reached the beginning of the train, he counted 120 steps. Then he immediately turned back and started walking in the opposite direction, again counting the steps from the beginning to the end of the sled train. As the train was moving toward him now, he only counted 40 steps before he reached it's end. Can he find out how long the train is and win the bet despite the kid's prank? (this puzzle is adapted from a book by C. Koval)
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4 comments:
Yay a puzzle with me in it! Very exciting :)
W says that the train is 120 feet.
He comes to this conclusion by assuming that his pace is 2 feet / sec and that the kids are pulling their sled train at some unknown S / sec.
W takes 120 sec to take 120 steps.
Let's call the length of the train T and the speed of the train S
On the first pass:
120 steps = 240 feet = T + 120*S
That is, W took 120 steps, 240 feet to pass the length of the train and the speed of the train * 120 seconds (the time it took for him to walk 120 steps)
On the second pass
40 steps = 80 feet = T - 40*S
That is, it took W 40 steps to walk 80 feet which was the length of the train minus the speed of the train in 40 seconds.
So we have:
240 = T + 120*S
80 = T - 40*S
We want to find T so we multiply the second equation by 3
240 = T + 120*S
240 = 3T - 120*S
We add the equations together, getting rid of S
480 = 4T
and then solve for T = 120.
So the length of the train is 120 feet!
W just beat me to it...
I also got 120 by doing:
B = rate of speed of boy
W = rate of speed of W
t1= time it takes for W to get to front of sled
t2= time it takes for W to return to end of sled
x = length of sled
W*t1 - B*t1 = x
W*t2 + B*t2 = x
W*t1 = 240 (120 steps times 2 feet per step)
W*t2 = 80
t1 = 3t2
W*3*t2 - B*3*t2 = W*t2 + B*t2
(algebra here...)
B*t2 = 1/2 * W*t2
B*t2 = 40
from 2nd equation above:
80 + 40 = 120
I like it how W. is writing in a third person about himself. Both solutions are correct, W. is again first and get a puzzle point.
This looks fascinatingly puzzling that W. and Phil submitted their solutions within minutes from each other three times! I need to investigate it.
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