Billiards Math!

Number Theory Level 2

On an $a \times b$ Billiards table, a Billiard ball starts going up and in right direction from its corner leftmost, lowermost corner. The ball has bounced from the top or bottom of the table $20$ times and from the right or left side of the table $12$ times Which direction is the ball now going?

This problem is not original, but a modified version.

Up and to the right direction Up and to the left direction Down and to the right direction Down and to the left direction

2 solutions

Matthew Christopher
Jul 27, 2020

If the ball makes an even number of bounces in a direction then its motion in that direction is unchanged from the start. For example, if I start going up then bounce off the top and bottom for a total of 2 times, I am still going up.

Since $20$ and $12$ are both even, the ball's motion is in the same direction as it started, so the answer is $\color{#20A900}{\boxed{\textnormal{up and to the right}}}$ .

Siddharth Chakravarty
Jul 27, 2020

The ball always has hit the upside of the table odd times, while the downside even times. As 20 is even, it means it has hit the downside, and is now travelling up. The ball will always hit the right side odd times and the left side even times. As 12 is even, it means it has hit the left side and is now travelling right. So combining both, it is going up-right direction.

Not posting why this is, use your brain a little, not writing everything.

Siddharth Chakravarty - 10 months, 2 weeks ago

I don't have brain, I think?

Vinayak Srivastava - 10 months, 2 weeks ago

Then how did you write this Number Theory Level 0 problem?

Siddharth Chakravarty - 10 months, 2 weeks ago

So that 64% people get it right!

Vinayak Srivastava - 10 months, 2 weeks ago

Billiards Math!

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