Dudeney puzzle - catching the thief
"Now constable", said the defendant's council in cross-examination, "you say that the prisoner was exactly 27 steps ahead of you when you started to run after him?" "yes sir" "and you swear that he takes 8 steps to your 5?" "that is so" "then i ask you, constable, as an intelligent man to explain how you ever caught him if that is the case?" "well you see, i have got a longer stride. In fact, 2 of my steps are equalin length to 5 of the prisoner's. If you work it out you will find that the number of steps I required, would bring me exactly to the spot where I captured him".
Here the foreman of the jury asked for a few minutes to figure out the number of steps the constable must have taken.
Let the distance where the defender and the prisoner met be D.
Then the distance the defender ran = D
Then, the distance the prisoner ran = D-27
The number of steps that the defender ran to D is x steps.
The number of steps that the prisoner ran to D is y steps.
2 steps of the defender = 5 steps of the prisoner.
==> 2x = 5 y
==> y= (2/5)x..............(1)
The defender run at a speed of 5 steps.
==> The defender's speed S1= 5x
The prisoner run at a speed of 8 steps.
The prisoner speed S2 = 8y
But y= (2/5)x
==> S2 = 8*(2/5)x = (16/5)x
Now we will determine the distance.
We will assume that the time is 1 unit.
==> D= S1 * T
==> D = 5x ..............(1)
Now we will determine the distance the prisoner ran,
==> D-27 = S2*T
==> D-27 = (16/5)x
==> D = (16/5) x + 27 ..............(2)
Now from (1) and (2).
==> 5x = (16/5)x + 27
==> 5x - (16/5)x = 27
==> (9/5)x = 27
==> x = 27*5/9 = 3*5 = 15
Then, the number of steps the defender ran is 15 steps.