Possible Solution to Johan Roos' River-Man problem

Let me suggest that we use one thread per riddle, otherwise it takes too much time to sort out too many ideas. :)

With all my due respect, let me re-state Johan's problem as follows, so that this thread is complete by itself:

A man fell and landed at an arbitrary spot in an infinitely long river of 100m width. It was pitch dark, the river flow was so slow that its flow speed was irrelevant. What is the shortest distance for the man to swim safely to the shore?

My solution is 200m. If others already posted the solution, let me apologize.

For easy explanation, let us assume that the river runs south/north bound. The man starts from a point near the west shore of the river. He swims northbound, parallely to the shore for 100m, (A -> B) then turn perpendicularly eastbound for another 100m. One can use trigonometry to prove that this route is the most unfortunate situation for the man.

If he swims slightly tiltedly toward northeast, i. e., A-> C, then

B

B' C

| /

|p/

|/ AB=AC=100m

A

The distance B'C = 100*sin(p), where p is the angle between AB and Ac. At C, he then makes a perpendicular right turn. When he intersects the other shore of the river, the point is called D. The distance CD can be expressed as

CD = 100*(1-sin(p))/cos(p).

And it can be proved that the number

(1-sin(p))/cos(p) is less than one if p < 90 degree. or CD is always less than 100m, or the original route stated above is the most unfortunate situation for the man.

If the man starts at point away from the shore, he will reach the shore in less than 200m. :)

If I understand you correctly your proposed path to swim is

straight 100m the turn right 90 deg then swim 100m

unfortunately this doesnt do the trick...

suppose he starts close to the west shore and gets his initial diection to be 45 deg from the shore going southwards...

good idea with starting a new thread :)

what if the man starts swimming (somewhere close to the shore) in a 45 degree angle away from the shore. then after 100m he turns 90 degrees and swims another 100 m. From what i see, he then ends up at the same distance from the shore as he started, just about 141m further down the river.

Hi, Johan and Michael, for the sake of clarity, let us start from a point near the WEST shore.

(a) If the man swims exactly toward northeast (p=45 degree) for 100m, and then makes a 90-degree right turn, he will travel exactly x to intersect the east shore:

x =100 + 100*(1-sin(p))/cos(p) = 100 + 41.4 = 141.4.

(b) If the man swims exactly toward east (p=90), clearly, he will travel only 100 to intersect the east shore.

(c) If 90 degrees < p < 135 degrees, yes, the man is in trouble. :)

OK, I was too rash and remiss. will try again. Thanks (for the riddle, and for the corrections)

OK, this ought to get me a Nobel prize or something equivalent. :) Let us call our solution x.

To continue our previous analysis, we find:

(a) if p < 90 degrees, then x < 200m.

(b) if 90 < p < 135, then, let me elaborate.

(c) if 135 < p < 180, then x < 200 again.

For case (b), the man will:

swim 100m straight --> make a 90-degree right turn --> swim for another 100m. At this point, if he sees he is still stuck in the middle of nowhere, then --> makes a 90-degree LEFT turn and continue to swim a distance of y, where

y = 100*(1/cos(q)-1) + 100*tan(q), q = p - 90.

For example, if q = 20, then y = 42.8m,

if q= 44.9, then y = 140.83m.

the worse scenario x = 100 + 100 + y = 340.8m.

So, my brute-force simple-minded solution depends on if the man was a good student in high school or not. :) It is important to pay attention to your trigonometry class, because doing so may save one's life.

I think you got a solution now but

you make the poor man swim a bit too long.

Nick Wedd has found (in the solutions thread) a very good solution, although even his can be bettered by a nudge ;)

Come to think of it, your last attempt doesn't really cut it either. Take your curve and try to fit it between the riverbanks by turning it around, if this is possible your solution does not cut it.

As the question stands the best solution is 50 m. Swim on the place until dawn, and then you swim to the closest shore.

javerberg, thats a true riddlelike answer :)

(guess Tims cornfield version has many advantages)

bump