Math Is Fun Forum

The answer is a numerical value, not in terms of p and q.

Totally wrong.

Label the letters like this

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Let's start with C1. There are 3 ways.

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By symmetry there are also 3 ways starting with C5, C6 and C10. This gives 12 ways so far.

Now let's start with C2.

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By symmetry, there are also 13 ways starting with C4, C7 and C9. So there are 52 ways, giving 64 ways so far.

Now let's start with C3. If the second letter is O1 or O2, we get 13 ways as if we start with C2. If we go C3O3, we get 3 ways as if we start with C5. This gives us 16 ways. By symmetry there are also 16 ways starting with C8. So there are 32 ways starting with the middle top or bottom C.

Hence there are exactly 96 ways to spell COOL.

Right. This means that in order to have a winner we must have one team winning two consecutive games from a position of parity. Hence an even number of games is required, so our sample space is {2, 4, 6, …}.

This problem is harder than I thought.

I took it that the winner was the first to win two games in a row. Is that correct?

NB: Notice that

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so I'm very confident my formula is correct.

Let the mass of the bug be m, the normal reaction by the surface on it be R, the radius of the ball be r, the velocity of the bug be v, and the angle with the vertical be θ.

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The centripetal force on the bug is given by

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When contact with the surface is lost, R = 0; then

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At that point the potential energy lost by dropping the vertical height is equal to the kinetic energy gained:

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[1] & [2] ⇒

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