Question: Where is the starting point for all balls?
About the article "Croquet", section "Garden".

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Packing in 3-dimensional containers.
Spheres into a Euclidean ball.
The problem of finding the smallest ball such that formula_1 disjoint open unit balls may be packed inside it has a simple and complete answer in formula_2-dimensional Euclidean space if formula_3, and in an infinite dimensional Hilbert space with no restrictions.
It is worth describing in detail here, to give a flavor of the general problem.
In this case, a configuration of formula_1 pairwise tangent unit balls is available.
Place the centers at the vertices formula_5 of a regular formula_6 dimensional simplex with edge 2; this is easily realized starting from an orthonormal basis.
A small computation shows that the distance of each vertex from the barycenter is formula_7.
Moreover, any other point of the space necessarily has a larger distance from "at least" one of the formula_8 vertices.
In terms of inclusions of balls, the formula_8 open unit balls centered at formula_10 are included in a ball of radius formula_11, which is minimal for this configuration.
To show that this configuration is optimal, let formula_12 be the centers of formula_8 disjoint open unit balls contained in a ball of radius formula_14 centered at a point formula_15.
Consider the map from the finite set formula_16 into formula_17 taking formula_18 in the corresponding formula_19 for each formula_20.
Since for all formula_21, formula_22 this map is 1-Lipschitz and by the Kirszbraun theorem it extends to a 1-Lipschitz map that is globally defined; in particular, there exists a point formula_23 such that for all formula_24 one has formula_25, so that also formula_26.
This shows that there are formula_8 disjoint unit open balls in a ball of radius formula_14 if and only if formula_29.
Notice that in an infinite dimensional Hilbert space this implies that there are infinitely many disjoint open unit balls inside a ball of radius formula_14 if and only if formula_31.
For instance, the unit balls centered at formula_32, where formula_33 is an orthonormal basis, are disjoint and included in a ball of radius formula_34 centered at the origin.
Moreover, for formula_35, the maximum number of disjoint open unit balls inside a ball of radius r is formula_36. 
The game of Chicago is played in a similar fashion to nine-ball and rotation, where balls must be played in order starting with the 1 ball.
In Chicago, all fifteen balls are used.
The money balls are the 1, 5, 8, 10, 13 and 15.
These six balls are racked to the front of the rack with the 1 ball at the head of the rack.
A player continues his turn as long as a ball is pocketed.
A score is kept as to how many money balls are made by each player.
Once all balls are pocketed the players will "settle up" based on the amount of money balls they made during the game.
The loser pays the winner a previously agreed upon amount multiplied by the difference in money balls between the two players (i.e., if Player A makes five money balls and Player B makes one then Player B owes Player A four times the amount of the wager.
)The game can also simply be played recreationally for points instead of money.
As well as the 1, 5, 8, 10, 13, and 15 balls being money balls (also called "Ways"), the number value of each ball pocketed by a player is added up at the end of each game (i.e.
1 ball = 1, 12 ball = 12).
The player(s) with at least 61 cumulative points, or a majority of points (the cumulative sum of all balls is 120) gets a 7th Way, or point.
This prevents the game from ending in a tie.
This game can be played by up to 4 players.
The 3 and 4 player format is team based, and decided by the following: 3 Players: The player who pockets the one ball is solo for the duration of the game, and competes against the other two players, who are now teammates, and shoot alternately against player one.
4 Players: The player who pockets the one ball remains alone until the lowest numbered money ball is pocketed by another player, making them teammates.
For example: player 1 pockets the one ball, and player two pockets the five ball.
They are now teammates, and competing against players three and four, who are also now teammates.
If player one pockets the one ball and the five ball, the player who pockets the 8 ball becomes teammate to player 1.
If the 1, 5 and 8 ball are pocketed by player one, the 10 ball then becomes the pairing ball, and so on.
Fouls: If you do not hit the object ball it is considered a push.
In the event of a push you have the option to either take the next shot or give it back to the person who pushed.
If there are two fouls in a row the next player in rotation gets ball in hand.
Scratches (Cue Ball hit into pocket) results in ball placed behind headstring.
Special considerations: Roundhouse: A roundhouse occurs if a player or team pockets all money balls on the table (1, 5, 8, 10, 13, 15).
This results in double points being awarded to that individual or team (or 2x the money per Way).
In this case the 61 points rule for the seventh Way does not apply.
Squarehouse: A squarehouse occurs if a player or team pockets ALL balls on the table, without an opponent pocketing a single ball.
This results in quadruple points being awarded to that individual or team (or 4x the money per Way). 
The impact produces a shock wave that propagates through the intermediate balls.
Any efficiently elastic material such as steel will do this as long as the kinetic energy is temporarily stored as potential energy in the compression of the material rather than being lost as heat.
Intrigue is provided by starting more than one ball in motion.
With two balls, exactly two balls on the opposite side swing out and back.
than half the balls can be set in motion.
For example, three out of five balls will result in the central ball swinging without any apparent interruption.
While the symmetry is satisfying, why does the initial ball (or balls) not bounce back instead of imparting nearly all the momentum and energy to the last ball (or balls)?
The simple equations used for the conservation of kinetic energy and conservation of momentum can show this is a "possible" solution, but they cannot be used to "predict" the final velocities when there are three or more balls in a cradle, because they provide only two equations to find the three or more unknowns (velocities of the balls).
They give an infinite number of possible solutions if the system of balls is not examined in more detail. 
Consider an urn containing balls of "K" different colors.
Initially, the urn contains "α"1 balls of color 1, "α"2 balls of color 2, and so on.
Now perform "N" draws from the urn, where after each draw, the ball is placed back into the urn with an additional ball of the same color.
In the limit as "N" approaches infinity, the proportions of different colored balls in the urn will be distributed as Dir("α"1...,"αK").
For a formal proof, note that the proportions of the different colored balls form a bounded [0,1]"K"-valued martingale, hence by the martingale convergence theorem, these proportions converge almost surely and in mean to a limiting random vector.
To see that this limiting vector has the above Dirichlet distribution, check that all mixed moments agree.
Note that each draw from the urn modifies the probability of drawing a ball of any one color from the urn in the future.
This modification diminishes with the number of draws, since the relative effect of adding a new ball to the urn diminishes as the urn accumulates increasing numbers of balls.
This "diminishing returns" effect can also help explain how small "α" values yield Dirichlet distributions with most of the probability mass concentrated around a single point on the simplex. 
A variation of four-ball is popular in Central Europe, especially in the Czech Republic.
It is called desítkový karambol (Czech for "tenfold carom") and is played with two white balls, a blue ball, and a red ball which serves as the cue ball.
Players score a point by hitting on any other two balls with the cue ball.
A hit off all three balls, however, scores 10 points, one point shot and 10 points shot is doubled by hitting a cushion before hitting any of the other balls for a total of 2 or 20 points in one shot.