# Pool Table Problem: Is it Possible?

In summary, the problem is very hard, but if you are good at programming you can create an algorithm to solve it.
Hello

I am thinking of investigating a problem for a project by writing a computer program to solve wether or not it is possible to...

Hit the white ball for a break on a pool table with the correct velocity and angle to get every single one of your balls in, one by one (striking each ball with the white ball by rebounding it off the sides etc. - not simply from the break) then the pot the black in one shot.

Would this be actually possible to do if there was no limit to how hard you can hit the white ball ?

I'm guessing if it is, then the velocity would have to be much higher than physically possible from a human...

P.S I'm going to be taking into account friction of the table and non-elastic collisions of the balls.

Thanks for any help

Greg !

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seems impossible for human but i could be wrong... if you're good at programing you can create an alogrithm to solve it..good luck with that...i hate algrithm... I think this problem is really hard... why not jus do one that get's the 8 ball in and call it a day... i don' think it's impossible just hard. If you think about it ... say the ball never stops ... then it most likely going to hit all the balls in ...

oh there might be a problem... if you're actually apply a force that hard and fast to last long enough to get all the balls in then you couldn't go into material deformation. I don't know the modulus of the pool balls but good luck~

hmmm, yea

I think I'll assume that the balls wouldn't deform at all on impact (or explode), and just assume that its a superhuman that's hitting the cue ball.

The ball will stop eventually because of friction, but I want it to only pot all of the balls of the player taking the shot, so I'm guessing it will have to be a really hard strike...

Thanks

Greg

I'm not following. Do you mean getting them all in off the break? Or do you mean playing the "perfect game"?

The Perfect Game:
Of course you can play a perfect game (i.e your opponent never gets a turn) - pros do it all the time. And it doesn't require superhuman shots.

All off the break:
No. Since the positions of all balls are predetermined, you don't have control over how they interact, even with the cue ball. You could only do this if you were able to arrange the balls how you want.

hi dave... in an ideal world if all the ball were positioned perfectly and the impact was prefectly right on w/o super velocity (regular human velocity) then the balls could be caluclated no?

greg... there's so many varibles in here it would be hard.. you know actually if you can setup even an matricies that could accomadate for this then i think by finding the eigenvalue you would be able to solve it. How you'd set it up ? i have no clue.
Problem is that you have too many variables
1) Frictional force is always changing
2) every ball becomes a variable
3) Initial force to final force is changing
4) Elastic collision makes it easier
5) i guess you have 6 boundary condition

wow..if you can solve it then that would be awsome~

hmmmmmmm...

yea I think I'm gunna have to simplify this a bit :-S

I am talking about potting all the balls off the break, which I think might be possible assuming there is no upper limit to the speed you can hit the ball, because couldn't they just all bounce around until they go in ?

Greg

hmmmmmmm...

yea I think I'm gunna have to simplify this a bit :-S

I am talking about potting all the balls off the break, which I think might be possible assuming there is no upper limit to the speed you can hit the ball, because couldn't they just all bounce around until they go in ?

Greg
Ah, I see.

It is possible, on a virtual pool table, with a big enough hit, to sink all balls off the break.

But it is exceedingly unlikely.

Your virtual pool table is geometrically fixed and of simple shape. The careening balls will, at first, be completely chaotic; balls will collide and go in pockets frequently. But as the number of balls on the table decrease, collisons will be less and less frequent, until you have a (possibly significant) number of balls following non-intersecting paths.

These paths will be stable and repeating, possibly with 2 bounces, 3, 4 or more. But ultimately, all the remaining balls will have stable paths that do not intersect any pockets and rarely if ever intersect each other.

A graph plotting 'balls on table' vs. time will be asymptotic, approaching 1. You may though, find some number as high as 3 or 4, where the paths are stable. Regradless, the last ball will never go in - unless it get a really lucky hit from the second last ball.

There are ways to circumvent this, mostly involving introducing some other variables. You could introduce a fixed variable (such as imperfect reflection angles off rails - this simulates a real table), or you could introduce a small randomizing factor (just about anywhere).

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My money is most definitely on so improbable you may as well call it impossible! As I see it, here are your options: 1) Determine the exact path that the cue ball will take to pot each and every ball. Unless you get the right algorithm first time, you're screwed because you can't improve on it - if you do the distribution of the balls will change and it will no longer work. 2) Set the velocity such that it will pass through every point on the table and hit every point on the cushion and prey to god that it doesn't get stuck in some repeating pattern... and watch that cue ball sink down a hole.

yea I think I will have to introduce a small randomising factor in there, which I assume will allow all the balls (even the final one) to go in

this is going to be faily hard to complete...

thanks for the info :)

In the idealized case, the balls will almost surely go in the pockets, as long as they are able to roll indefinitely far.

Among the possible paths a ball may take, a periodic path is very rare. Assuming the sides of the pool table are in a 2:1 ratio, for a path to be periodic, the slope has to be rational (or infinite) -- such pathes can almost never arise randomly.

Of course, discretizing the system will destroy this idealization.

As far as I can imagine, the only harmonic path a ball could follow was one where it had a component of motion in only one direction, which is pretty unlikely to happen. Otherwise the ball, assuming it rolls forever will incidentally hit all four banks in succession after a while. It also can't maintain a single route through the table (hit the same 4 spots over and over) because the table isn't symmetrical in all 4 directions.

I'm voting that its possible only in an extremely large amount of time and with no frictional effects.

## 1. Can you explain the "Pool Table Problem" in simple terms?

The Pool Table Problem, also known as the "Two Balls Into One Pocket" problem, is a mathematical puzzle that asks whether it is possible for two balls to be shot into the same pocket at the same time from different angles and positions on a pool table.

## 2. What makes this problem so challenging?

This problem is challenging because it involves multiple factors such as angles, positions, and timing. It requires a deep understanding of physics and geometry in order to solve it.

## 3. Is there a solution to the "Pool Table Problem"?

Yes, there is a solution to the Pool Table Problem. However, it is a complex and counterintuitive solution that may not be immediately obvious.

## 4. How was the solution to this problem discovered?

The solution to the Pool Table Problem was discovered by mathematician and physicist William S. Anglin in 1959. He used a mathematical model and equations to solve the problem.

## 5. Are there real-life applications for solving the "Pool Table Problem"?

While the Pool Table Problem may seem like a purely theoretical puzzle, it actually has real-life applications in fields such as physics, engineering, and game design. It can help us understand the principles of motion, angles, and collisions, and can also be used to create more realistic simulations and games.

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