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Vandaag is het 21 January 2018

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Beach walk - math puzzle

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Along the beach are a number of poles standing at equal distance from each other. The poles are numbered 1, 2, 3, 4, ..... Alice is walking from the first pole to the last one and back, Bob is doing that in the opposite direction. They start at the same time and walk with constant, but different velocity. Their first encounter is at pole number 10, their second (when they're both on the way back) at pole number 20. How many poles are standing along the beach?
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Explanation

The distance between the poles is not important, take it for example 100 m. Let us call the distance between the first and the last pole D. At the first encounter, Alice and Bob have together walked a distance D. At the second encounter (as one can see by making a sketch) they have together walked 3 times as much, so 3D. At the first encounter Alice has walked 900 m, so at the second encounter she has walked 2700 m. Then she still has to go back 1900 m to her start at pole 1. So in total Alice walked 4600 m. From this it follows that D = 2300 m, so there are 24 poles along the beach.

This puzzle can also be solved in a quite different less elegant way. Let us call the velocity of Alice v_a and that of Bob v_b and the number of poles x. At the first encounter at time t_1 it holds that
t_1 = 9 / v_a = (x - 10) / v_b.
At the second encounter at time t_2 it holds that
t_2 = (x - 1) / v_a + (x - 20) / v_a = (x - 1) / v_b + 19 / v_b.
These equations can be rewritten into
9 v_b = (x - 10) v_a
(2x - 21) v_b = (x + 18) v_a
If we divide the upper equations by each other and multiply by the denominators we obtain the following quadratic equation
x^2 - 25 x + 24 = 0.
This equation has as solution x = 1 and x = 24. x = 1 is not a correct solution, since there are at least 20 poles along the beach. So we can conclude that there are 24 poles along the beach.