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Billiard Architect

Billiard Architect

AzB Silver Member
Silver Member
I was thinking about this the other day and figured it would be interesting to know...

What are the possible number of layouts that a pool table can have? So I figured it out...

This is for a 9 foot table and regular 2 1/4 inch pool balls. I figured that 1/16th of an inch was sufficient divisions of the placement of a pool ball. In other words within an inch a ball can be placed @ 16 different points within that inch. Within a square inch a pool ball can rest 16 x 16 = 256 places within that square inch. So using that as a measurement.

First the constants.

Length of playing surface: 100 inches
Width of playing surface: 50 inches
Length of playing surface minus the 1 1/8 inch that the ball can not be placed because when a ball is against the rail the contact of the table is 1 1/8 inch from the rail. So 1 1/8 on one end plus 1 1/8 on the other end equals 2 1/4. Subtract that from the length: 97.75
Width of playing surface minus the 2 1/4 gap: 47.75
Length of table x 16 possible points per inch = 1564
Width of table x 16 possible points per inch = 764
Total possible ball placements L x W: 1,194,869

Now the variables.

Now it would be real easy just to take that number and power it times the number of balls on the table to come up with the number of possible layouts. But rember that no two balls can occupy the same space. and that diameter equals the 4.5 or 2x the size of a ball. so to find the area that a ball would occupy you square the radius and multiply it by pi (3.14). So take 2.25 (radius of area) times 16 to come up with the possible points along the radius. That equals 36. Then multiply that by itself to square the number. That equals 1,296. Then multiply that by 3.14 and come up with 4069.44. That is the number of places out of the 1,194,869 (total ball placements) that a ball cannot be placed on the table if another ball was there. Now comes the easy part. You take the number of balls on the table times the area that is occupied by a ball on the table and subtract that from the total possible ball placements. Take the answer and power it to the number of balls on the table. (don't forget the cue ball) Here is the possible patterns that I came up with for some of our common games.

Straight pool, 8 Ball, One Pocket (15 balls + 1 cueball): 7.46E+96
9 Ball (9 balls + 1 cueball): 4.35E+60
7 Ball (7 balls + 1 cueball): 3.43E+48
3 ball (3 balls + 1 cueball): 1.96E+24

So when you break the balls tonight and nothing goes you can relish in the thought that you have a 4.35E+60 chances to see that pattern ever again.

JV
 
This reminds me of a proposition shot. Bet someone that they cannot make a shot that you will make and you will give them 100 tries. Rack the balls and then break them. Mark where all the balls are and now it is their turn to duplicate it.
 
Johnny "V" said:
I was thinking about this the other day and figured it would be interesting to know...

What are the possible number of layouts that a pool table can have? So I figured it out...

This is for a 9 foot table and regular 2 1/4 inch pool balls. I figured that 1/16th of an inch was sufficient divisions of the placement of a pool ball. In other words within an inch a ball can be placed @ 16 different points within that inch. Within a square inch a pool ball can rest 16 x 16 = 256 places within that square inch. So using that as a measurement.

First the constants.

Length of playing surface: 100 inches
Width of playing surface: 50 inches
Length of playing surface minus the 1 1/8 inch that the ball can not be placed because when a ball is against the rail the contact of the table is 1 1/8 inch from the rail. So 1 1/8 on one end plus 1 1/8 on the other end equals 2 1/4. Subtract that from the length: 97.75
Width of playing surface minus the 2 1/4 gap: 47.75
Length of table x 16 possible points per inch = 1564
Width of table x 16 possible points per inch = 764
Total possible ball placements L x W: 1,194,869

Now the variables.

Now it would be real easy just to take that number and power it times the number of balls on the table to come up with the number of possible layouts. But rember that no two balls can occupy the same space. and that diameter equals the 4.5 or 2x the size of a ball. so to find the area that a ball would occupy you square the radius and multiply it by pi (3.14). So take 2.25 (radius of area) times 16 to come up with the possible points along the radius. That equals 36. Then multiply that by itself to square the number. That equals 1,296. Then multiply that by 3.14 and come up with 4069.44. That is the number of places out of the 1,194,869 (total ball placements) that a ball cannot be placed on the table if another ball was there. Now comes the easy part. You take the number of balls on the table times the area that is occupied by a ball on the table and subtract that from the total possible ball placements. Take the answer and power it to the number of balls on the table. (don't forget the cue ball) Here is the possible patterns that I came up with for some of our common games.

Straight pool, 8 Ball, One Pocket (15 balls + 1 cueball): 7.46E+96
9 Ball (9 balls + 1 cueball): 4.35E+60
7 Ball (7 balls + 1 cueball): 3.43E+48
3 ball (3 balls + 1 cueball): 1.96E+24

So when you break the balls tonight and nothing goes you can relish in the thought that you have a 4.35E+60 chances to see that pattern ever again.

JV
Click to expand...


Did you get a gov grant for this?
 
Say what???

Johnny "V" said:
I was thinking about this the other day and figured it would be interesting to know...

What are the possible number of layouts that a pool table can have? So I figured it out...

This is for a 9 foot table and regular 2 1/4 inch pool balls. I figured that 1/16th of an inch was sufficient divisions of the placement of a pool ball. In other words within an inch a ball can be placed @ 16 different points within that inch. Within a square inch a pool ball can rest 16 x 16 = 256 places within that square inch. So using that as a measurement.

First the constants.

Length of playing surface: 100 inches
Width of playing surface: 50 inches
Length of playing surface minus the 1 1/8 inch that the ball can not be placed because when a ball is against the rail the contact of the table is 1 1/8 inch from the rail. So 1 1/8 on one end plus 1 1/8 on the other end equals 2 1/4. Subtract that from the length: 97.75
Width of playing surface minus the 2 1/4 gap: 47.75
Length of table x 16 possible points per inch = 1564
Width of table x 16 possible points per inch = 764
Total possible ball placements L x W: 1,194,869

Now the variables.

Now it would be real easy just to take that number and power it times the number of balls on the table to come up with the number of possible layouts. But rember that no two balls can occupy the same space. and that diameter equals the 4.5 or 2x the size of a ball. so to find the area that a ball would occupy you square the radius and multiply it by pi (3.14). So take 2.25 (radius of area) times 16 to come up with the possible points along the radius. That equals 36. Then multiply that by itself to square the number. That equals 1,296. Then multiply that by 3.14 and come up with 4069.44. That is the number of places out of the 1,194,869 (total ball placements) that a ball cannot be placed on the table if another ball was there. Now comes the easy part. You take the number of balls on the table times the area that is occupied by a ball on the table and subtract that from the total possible ball placements. Take the answer and power it to the number of balls on the table. (don't forget the cue ball) Here is the possible patterns that I came up with for some of our common games.

Straight pool, 8 Ball, One Pocket (15 balls + 1 cueball): 7.46E+96
9 Ball (9 balls + 1 cueball): 4.35E+60
7 Ball (7 balls + 1 cueball): 3.43E+48
3 ball (3 balls + 1 cueball): 1.96E+24

So when you break the balls tonight and nothing goes you can relish in the thought that you have a 4.35E+60 chances to see that pattern ever again.

JV
Click to expand...


Johnny,,,,,my man,,,,,,I would have to have one of those flash-backs to to the glory days of, Wonderful Window Pains, to follow you on this one.
 
Did you know....................the toothbrush was invented in Arkansas?

If it would've been invented anywhere else....they would've called it a teethbrush.
 
Einstein equated that there were over 5 million shot posibilities in a 9ft. table....randyg
 
You made my head hurt, but oddly enough, I followed your reasoning. :p

Now you just have to figure out how many shots there are possible in a game relative to the cueball's position, because that would include all the balls plus a diminishing number of balls after each shot.

Then when you are done, you can figure out how many patterns there are to get to the next ball for each configuration, taking into account average human strength. In other words, you can't go an infinite number of rails.
 
Johnny "V" said:
I was thinking about this the other day and figured it would be interesting to know...

What are the possible number of layouts that a pool table can have? So I figured it out...

This is for a 9 foot table and regular 2 1/4 inch pool balls. I figured that 1/16th of an inch was sufficient divisions of the placement of a pool ball. In other words within an inch a ball can be placed @ 16 different points within that inch. Within a square inch a pool ball can rest 16 x 16 = 256 places within that square inch. So using that as a measurement.

First the constants.

Length of playing surface: 100 inches
Width of playing surface: 50 inches
Length of playing surface minus the 1 1/8 inch that the ball can not be placed because when a ball is against the rail the contact of the table is 1 1/8 inch from the rail. So 1 1/8 on one end plus 1 1/8 on the other end equals 2 1/4. Subtract that from the length: 97.75
Width of playing surface minus the 2 1/4 gap: 47.75
Length of table x 16 possible points per inch = 1564
Width of table x 16 possible points per inch = 764
Total possible ball placements L x W: 1,194,869

Now the variables.

Now it would be real easy just to take that number and power it times the number of balls on the table to come up with the number of possible layouts. But rember that no two balls can occupy the same space. and that diameter equals the 4.5 or 2x the size of a ball. so to find the area that a ball would occupy you square the radius and multiply it by pi (3.14). So take 2.25 (radius of area) times 16 to come up with the possible points along the radius. That equals 36. Then multiply that by itself to square the number. That equals 1,296. Then multiply that by 3.14 and come up with 4069.44. That is the number of places out of the 1,194,869 (total ball placements) that a ball cannot be placed on the table if another ball was there. Now comes the easy part. You take the number of balls on the table times the area that is occupied by a ball on the table and subtract that from the total possible ball placements. Take the answer and power it to the number of balls on the table. (don't forget the cue ball) Here is the possible patterns that I came up with for some of our common games.

Straight pool, 8 Ball, One Pocket (15 balls + 1 cueball): 7.46E+96
9 Ball (9 balls + 1 cueball): 4.35E+60
7 Ball (7 balls + 1 cueball): 3.43E+48
3 ball (3 balls + 1 cueball): 1.96E+24

So when you break the balls tonight and nothing goes you can relish in the thought that you have a 4.35E+60 chances to see that pattern ever again.

JV
Click to expand...

Man, this is way above my head!

bunnync3.jpg

I forget who I stole this picture from, but I've been dying for an opportunity to use it!
 
42NateBaller said:
Man, this is way above my head!

View attachment 40041

I forget who I stole this picture from, but I've been dying for an opportunity to use it!
Click to expand...

This is fantastic! I am stealing it from you, and will now begin waiting for a similar opportunity to use it. !
 
there are about 20 different pictures of that rabbit on this forum in one thread....i dont remember the thread but zomb started it and we traded bunny pictures for about two days.... wiht the grand finale being a bunny with pancakes on hi head balacned on a cup cake on another bunnys head...or something like that


very famous bunny there
 
By any chance, before you started all these calculations, did a ball fly off a table and hit you in the head? :cool:
 
acedotcom said:
By any chance, before you started all these calculations, did a ball fly off a table and hit you in the head? :cool:
Click to expand...
I just got tired of people saying that there are an infinitesmal amount of patterns on a table... No... Its pretty high up there but it certainly is a finite number. :)

Man that bunny was hilarious!!!
 
Your calculations are incorrect. I will consult some books on combinatorics and get back here with a valid solution. Your numbers are far too low.
 
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Johnny "V" said:
I just got tired of people saying that there are an infinitesmal amount of patterns on a table... No... Its pretty high up there but it certainly is a finite number. :)
Click to expand...
There are finite numbers and then there are finite numbers.

For a 'rough' sense of how big your numbers are, the number of molecules in a teaspoon of water is something like 10E+27 (I've forgotten the exact figure). If you spread this much water (the exact figure) evenly over the entire United States, including of course Alaska and Hawaii, you would count about 11 million molecules for every square inch of surface area.

But this number is almost infinitesimal compared to your E+60.

Jim

PS: We must have had a run in with the same ball!
 
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Johnny "V" said:
I just got tired of people saying that there are an infinitesmal amount of patterns on a table... No... Its pretty high up there but it certainly is a finite number. :)

Man that bunny was hilarious!!!
Click to expand...


i politely disagree
 
BazookaJoe said:
D'you know that my next door neighbor has three rabbits?
Click to expand...

Way cool.

Were you aware that they call those little, round, dark things under the cages, "Smart Pills"?

Ken
 
Johnny "V" said:
I was thinking about this the other day and figured it would be interesting to know...

What are the possible number of layouts that a pool table can have? So I figured it out...

This is for a 9 foot table and regular 2 1/4 inch pool balls. I figured that 1/16th of an inch was sufficient divisions of the placement of a pool ball. In other words within an inch a ball can be placed @ 16 different points within that inch. Within a square inch a pool ball can rest 16 x 16 = 256 places within that square inch. So using that as a measurement.

First the constants.

Length of playing surface: 100 inches
Width of playing surface: 50 inches
Length of playing surface minus the 1 1/8 inch that the ball can not be placed because when a ball is against the rail the contact of the table is 1 1/8 inch from the rail. So 1 1/8 on one end plus 1 1/8 on the other end equals 2 1/4. Subtract that from the length: 97.75
Width of playing surface minus the 2 1/4 gap: 47.75
Length of table x 16 possible points per inch = 1564
Width of table x 16 possible points per inch = 764
Total possible ball placements L x W: 1,194,869

Now the variables.

Now it would be real easy just to take that number and power it times the number of balls on the table to come up with the number of possible layouts. But rember that no two balls can occupy the same space. and that diameter equals the 4.5 or 2x the size of a ball. so to find the area that a ball would occupy you square the radius and multiply it by pi (3.14). So take 2.25 (radius of area) times 16 to come up with the possible points along the radius. That equals 36. Then multiply that by itself to square the number. That equals 1,296. Then multiply that by 3.14 and come up with 4069.44. That is the number of places out of the 1,194,869 (total ball placements) that a ball cannot be placed on the table if another ball was there. Now comes the easy part. You take the number of balls on the table times the area that is occupied by a ball on the table and subtract that from the total possible ball placements. Take the answer and power it to the number of balls on the table. (don't forget the cue ball) Here is the possible patterns that I came up with for some of our common games.

Straight pool, 8 Ball, One Pocket (15 balls + 1 cueball): 7.46E+96
9 Ball (9 balls + 1 cueball): 4.35E+60
7 Ball (7 balls + 1 cueball): 3.43E+48
3 ball (3 balls + 1 cueball): 1.96E+24

So when you break the balls tonight and nothing goes you can relish in the thought that you have a 4.35E+60 chances to see that pattern ever again.

JV
Click to expand...

NOT EVEN CLOSE!

Just for nine-ball (10 unique balls), the number of possible combinations of layouts on a 4.5'x9' pool table is larger than the amount of subatomic particles in the universe. The flaw in your calculations is that you are treating all the balls the same, but each ball is unique. To put it in English, you are not dealing with a pool table layout, but simply a pattern of dots on a grid. Since each ball is unique, the calculation is much more complex. This type of problem can only be figured out (approximately), by a faction of mathematics called, Combinational Analysis.

Since we will very soon be dealing with enormous numbers, let's simplify a few things. Let's not worry about the effect of the rails or other balls have on each other, and let's reduce the number of ball positions on the table. To do this, let's reduce the resolution of the grid from 1/16" to ball-width (2.25"). This reduces the number of positions a single ball can occupy on the table from 1.2 million, to 1152.

EDIT: I screwed up the math here somewhat. I'll leave it for everyone to see how dumb I was, but the correct calculation is below.

This means a single ball can occupy one of 1152 spots on our regulation pool table, so for one ball, the odds of it landing in any position is 1 in 1152. Now let's add a second ball. We will call the balls, ball 1 and, ball 2. Imaging our two balls laying in the first two slots in the top left corner of the table. This is one possibility. Let's keep the first ball in it's slot, and move the second ball to slot three. This is possibility number two, and so on and so forth until we have ball 1 in slot 1 and ball two in slot 1152. This is 1151 possible outcomes with ball 1 in slot 1. We are done with the first iteration of this calculation. Now lets move ball 1 to slot 2, and ball two to slot one; this is possibility 1152. We can't move ball 2 to slot 2 because it is occupied by ball 1, which is where ball 1 will remain for this iteration of the calculation, so we move it to slot 3, and so on and so forth until ball 2 is in slot 1152. We have gone through only two iterations of this calculation, and we now have 2302 possible combinations of only two balls. Obviously I am not going to take us all the way through all these 1152 iterations, my fingers already hurt! Luckily there is a mathematical shortcut (and a necessary one once we have more than two balls), called the Factorial (represented by !). For only 2 balls, the amount of possible combinations is 1152^2!, or 1327104. This happens to be 1152 squared because 2! = 2. Now it gets fun when you add the third ball. 1152^3! = 2337302235907620864, or 2.3x10^17 Already we are approaching the numbers you have proposed for 4 balls, but with 1.2 million possible positions, but in my example we only have 1152 possible positions. Let’s add ball 4. 1152^4! = 1152^3^74. Already we arrive at numbers that can not be expressed by writing them out, (unless you have a piece of paper the size of the solar system and a ton of time). So let’s skip right to 10 balls. That is 1152^10! Or 1152^3628800. That one breaks my calculator!


Now since we have a better grasp of how these calculations are done, we can go back to your original table resolution of approximately 1.2MM slots, and express the probability of a single table layout. This can only be done symbolically though, as the numbers become mind boggling very quickly. This number in itself is hard to grasp, but let me shed some light on it. There are only 1.6x10^23 stars in the UNIVERSE. This can be represented as 160000000000000000000000. 1152^10! is a number several trillion orders of magnitude larger than this.

If we go back to your table with a resolution of 1.2MM, then the expression for the number of possible layouts is1200000^10! This is how many possible layouts you can get in a game of nine-ball. It is a number beyond comprehension. The odds that the subatomic structure of your body were to suddenly resonate at exactly the correct frequency and allow you to simply fall through the subatomic structure of the Earth are far smaller than the odds you will ever see the exact same layout in nine-ball to a sixteenth of an inch resolution.

...going by this math.


BUT! there are some things we still have not accounted for. Here are a few I thought of

Pocketing balls. Eww! I'm not even getting into how that complicates the math.

The fact that when you break a rack the balls do not occupy the same places in the rack each time. Beyond calculation!

The fact that when you break, the odds of a ball ending up near the rack are greater than ending up at the head of the table because of the distance. This factor is actually going to lower the amount of possible layouts, but not by an amount that will make any difference to the size of the numbers involved; half of a huge number is still a huge number.

Because of the shape of the rack and position of the balls in it, certain balls are more likely to travel in certain directions, and have greater tendency to end up in certain locales.

Wow! Thanks for thinking of this though. It was a very interesting and fun thought experiment!

EDIT EDIT EDIT!!!

WHOOPS!

I used the wrong equation to calculate this. The correct equation can be found at: http://en.wikipedia.org/wiki/Combinatorics under "Combination with repetition".

The correct equation yields a similar number though. For a pool table with a resolution of 1.2MM slots a ball can occupy, the number of distinct patterns of balls is on the order of 1x10^10,000,000, or 1 followed by 10,000,000 zeros.
 
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