Puzzles with numbers and things

[elvaago]

Yeah, that was an easy one, wasn't it? But don't put yourself down, there's two more easy ones, in my opinion. It's the one hard one that has me totally stumped.

[elvaago]

One more comment. He does not use spin.

[snookersfun]

Yeah, that was an easy one, wasn't it? But don't put yourself down, there's two more easy ones, in my opinion. It's the one hard one that has me totally stumped.

OK, I'll try
first one: 100% chance of being potted at your crazy speed
last one: wouldn't that lead to infinity cushions, as Final speed=initial speed *(1-0.01)^n

[elvaago]

I agree with your two answers, though someone who purposeful misread my questions might offer the answer '4' to the maximum amount of cushions it can hit since there's only 4 cushions on a snooker table! But that was not my intention when I wrote the question, so perhaps I should say: How many times will it hit a cushion, rather than How many cushions are hit?

:-)

[davis_greatest]

OK, I'll try
first one: 100% chance of being potted at your crazy speed

Not sure about this. There might be a route where the white keeps going around the table (possibly eventually retracing its steps) without ever entering a pocket. Not sure.

last one: wouldn't that lead to infinity cushions, as Final speed=initial speed *(1-0.01)^n

Again, not sure. There may be no such route where the cue ball does not enter a pocket after a finite number of bounces. Haven't thought about it yet to decide!

The expected number of cushions question would take a bit of work, but is do-able (just take the average of the number of cushions hit for angles from 1 degree to 89 degrees). Too much effort for me though

[davis_greatest]

i like it d_g

Nice suggestion, semih You're already in the Ball of Frame.

[davis_greatest]

- What are the chances of the white ball being potted?

Less than 1 (but I don't have the patience to calculate it exactly). Vlad could play the white to hit the cushion at 45 degrees midway between the centre and corner pocket. It would then hit the centre of the bottom (or top) cushion, and then off the other side cushion at 45 degrees, pass through the blue spot, and so it would continue...

- How many cushions would you expect it to hit?

The above example would hit an infinite number of cushions and has a probability of greater than zero (since you said that Vlad can only hit the ball at an exact number of degrees, and I assume that each angle is equally likely). Therefore the expected number of cushions hit is infinite (even though the probability of this may be as low as 1 in 89).

- What is the least amount of cushions it has to hit?

One unless he doesn't hit the ball hard enough to reach a cushion - it depends how high "high speed" is for a thumbless doggy!

- What is the maximum amount of cushions it can hit?

Infinite, per example above - if we ignore friction.

[davis_greatest]

Round 105 - Oliver strikes back

Tonight, Oliver has also just made his first total clearance! (15 reds, each with a colour, and then the six colours)

I noticed that he likes potting the pink as much as the brown and he potted each of these exactly the same number of times. He also potted the same number of blues as greens; but 2 pots before each green, he potted a black. Actually, the number of points he scored from blacks was the same as the number of points he scored from yellows.

What was his break? (I'm a little tired and have done this in quite a hurry - Oliver only made the break a minute ago - so hopefully it still makes sense)

[snookersfun]

- What is the least amount of cushions it has to hit?

One unless he doesn't hit the ball hard enough to reach a cushion - it depends how high "high speed" is for a thumbless doggy!

In Elvaago's problem you would always hit that one cushion (could take a long time to reach, but it would), as the ball only looses speed on contact with a cushion and according to answer 4, balls only stop moving when in the pocket. I had that one thought out properly

[snookersfun]

Tonight, Oliver has also just made his first total clearance! (15 reds, each with a colour, and then the six colours)

I noticed that he likes potting the pink as much as the brown and he potted each of these exactly the same number of times. He also potted the same number of blues as greens; but 2 pots before each green, he potted a black. Actually, the number of points he scored from blacks was the same as the number of points he scored from yellows.

What was his break? (I'm a little tired and have done this in quite a hurry - Oliver only made the break a minute ago - so hopefully it still makes sense)

Assuming that the 'green-black' is mutually exclusive, i.e. 2 pots before every green a black has to be potted and therefore each black has to be followed by a green (except the last one):

a 101 total clearance

[snookersfun]

...just noticed, I don't need that assumption. The wording in the question is quite enough.

So, I'll also take this opportunity to tell steveb72: Welcome to the club! Well done and enjoy!

[elvaago]

There's a total of 21 instances of potting a coloured ball.

He scores as many points from the black as the yellow. He can't pot one black, because 7 is not dividable by 2. He can't pot 3 blacks either, because 21 is not dividable by 2. If he pots 4 blacks, his point total is 28, which means he would have to pot 14 yellows. That makes 18 balls potted. Since it's a total clearance, he still has to pot green, brown, blue and pink, which is 4 balls, making it a total of 22, which is not possible. So he has to have potted 2 blacks for 14 points and therefore 7 yellows also for 14 points.

He pots a black before every green. Since he can't pot a green ball after the last black, it's only possible that he pots the black ball with his last red ball, and therefore he only pots one green ball. He also pots one blue ball.
2 blacks + 7 yellows + 1 green + 1 blue = 11 balls. That leaves 10 balls for the pink and brown. He pots each the same amount as the other, so that means he pots each of them 5 times.
15 reds + 7 yellows + 1 green + 5 browns + 1 blue + 5 pinks + 2 blacks =
15 + 14 + 3 + 20 + 5 + 30 + 14 = a 101 point total clearance.

[davis_greatest]

In Elvaago's problem you would always hit that one cushion (could take a long time to reach, but it would), as the ball only looses speed on contact with a cushion and according to answer 4, balls only stop moving when in the pocket. I had that one thought out properly

I had thought about that, but that is not exactly what elvaago said. He said the ball loses speed when it hits a cushion. He did not say that the ball does not lose speed in between cushions, i.e. he did not say that the only time the ball loses speed is on contacting a cushion (but I assumed this was intended).

[davis_greatest]

Well done snookersfun and elvaago! 101 is correct.

Note that I said that 2 pots before each green, Oliver pots a black. I did not say that after every black, Oliver pots a green 2 pots later. That would be impossible, because after the final black there are no more pots!

[elvaago]

That's precisely the reason why he can only pot one green. :-)