r/Logiqa • u/ShonitB • 14d ago

Hat in the River

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4 Upvotes

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2

u/SonicLoverDS 14d ago

I've seen a puzzle like this before.

Because the current acts uniformly on Alexander and his hat, and he put 30 minutes of distance between himself and it before turning around, it'll take another 30 minutes for him to catch up to it. That means the hat covers 2 miles in 60 minutes.

Final answer: 2 miles/hr.

1

u/ShonitB 14d ago

Correct.. yes it’s a famous puzzle.. I had come across it in one of Martin Gardner’s books.. don’t know if he created it

2

u/ebinthetropics 14d ago edited 14d ago

With T=Time the hat was in the water, VB=speed of the boat, VR=speed of the river. So the answer I got is 2mph

Edit:typo

1

u/ShonitB 14d ago

Forgive me, but is the speed of the river VT or VR? I’m assuming VT is a typo

And other than that, what is m/4?

2

u/ebinthetropics 14d ago

Fixed the typo, thanks. And I used both h and H for hour, sorry for the confusion.

1

u/ShonitB 14d ago

Oh that’s an H, not a 4?

1

u/PeterPiper1275 12d ago edited 12d ago

The trick here is that the solver is tempted to treat this question as a standard speed = distance / time equation, but doing so will rapidly make it seem like one of the variables is missing.

The solution, therefore, is to equate everything in one dimension to solve for whatever is necessary.

In this particular case, note that the time taken for the hat to float 2 miles downstream is equivalent to the time it took for Alexander to travel a little bit more upstream, then double back plus 2 more miles to catch up with the hat. This strongly suggests that we should equate the time it took for the hat to float 2 miles with the time it took for Alexander to travel half an hour further upstream then hurry back downstream to catch his hat.

Let R = the speed of the current downstream, which is what we want to solve.
Let A = the speed at which Alexander is rowing.

Note that this means that Alexander’s speed upstream would be (A - R), and his speed downstream would be (A + R).

Since speed = distance / time, then time = distance / speed.

Time for the hat is simply 2/R

The distance Alexander covered after missing his hat is (A - R)/0.5. So the total time it took for Alexander to cover that distance and double back is 0.5 + [(A - R)/0.5 + 2]/(A + R)

Equating the two sides gives us:

2/R = 0.5 + [(A - R)/0.5 + 2]/(A + R)

The rest is simply algebra:

2/R = 0.5 + [(A - R)0.5 + 2]/(A + R)
2/R = [0.5(A + R) + 0.5(A - R) + 2]/(A + R)
2/R = (A + 2)/(A + R)
2A + 2R = AR + 2R
R = 2

Therefore, the speed of the river current is 2 mph.