Tick-Tock Triangles

Originally posted by The Plumber
What happens if the minute hand and hour hand move continuously, and not in discrete increments?

In fact, that was the intended meaning of the puzzle.

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In that case, I believe both your probabilities are zero. There are an infinite number of positions of the hands (because they move smoothly) and a finite number (c) of those hand positions will be isoceles or right angled triangles, therefore the probability that the clock is in one of those positions when you look at it is

c/infinity = 0.

Originally posted by iamatiger
In that case, I believe both your probabilities are zero. There are an infinite number of positions of the hands (because they move smoothly) and a finite number (c) of those hand positions will be isoceles or right angled triangles, therefore the probability that the clock is in one of those positions when you look at it is

c/infinity = 0.

Except that THUDandBLUNDER said, 'given that the triangle has integer length sides', so there are problems defining the relevant random variables.

Acolyte, your answer for 3rd part is partly right.

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Originally posted by THUDandBLUNDER

Acolyte, your answer for 3rd part is partly right.

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Well, the numerator is 2 🙂

Anyway, parts (i) and (ii) are easy.

Here's a good one:
Given a normal clock, where the second hand moves in full seconds, the minute hand in 60/th of a minute etc. If the hour hand is 3 cm long, and the minute hand is 4 cm long, at how many times during 12 hours are the tips of the hands separated exactly by an integral number of cm?