Finding the Number of Games.—The first thing to know is the number of games imposed by differing numbers of entries. There are several ways of working this out. The quickest and simplest is a mental one. If the number of entries is even, say 10, multiply the second highest term (9) by one-half the highest—9×5=45. If odd, as 7, multiply that figure by one-half the next highest—7×3=21 games. But, if a pencil is handy, a quick enough way is to multiply the highest term, whether odd or even, by the next highest, and then divide the product by 2, which will show 253 games if there are 23 entries—23×22 506, halved. These are regular games. No amount of figuring can forecast ties.

Tie Games.—When competition is for a championship, any tie for it must be played off. Ties for other prizes may or may not be, as circumstances dictate.

Tie Games Separate.—Save when they involve a championship, tie games are no part of the tournament proper, which ordinarily ends when all the contestants have either played or forfeited an equal number of games. Yet, while tie games for else than the championship will not serve to determine other than special wagers, they are, nevertheless, records in themselves, although without being a part of the tournament.

Forfeitures.—In a tournament, every game begins with the first one, in the sense of binding every player who has not previously been declared out. It has always been an unwritten law of billiards that a withdrawer, instead of canceling his games already played, forfeits those he has yet to play. The former procedure penalizes the faultless for another’s fault. It is also open to the objection that, in order to deprive one winner of his record for high average or high run, the loser of the game in which either was made may be induced to withdraw. Injustice is possible even without collusion. Within two years, cancellation has deprived one continuing player of his highest average, and another of the highest average of all.

For amateur tournaments, a few Western roomkeepers have a rule of their own, which cancels if the withdrawer has not played more than half his games, and forfeits if he has.

Guarding Against Forfeiture or Other Failure.—Until a scheduled game is started, the players of the next one in order should be on hand.

The Sanctity of Schedule.—A schedule once made out by due authority should be adhered to, instead of being changed to suit some individual caprice.

Opening Game.—Never let it be between the supposed best two players. For some special reason, one such may be utilized, but not two without inviting the almost certain penalty of a loss of public interest as the games draw near their close.

Rush the Losers.—As far as practicable, play losers first in preference to winners. If they are good losers, they will not object. In no other way can the anti-climax be prevented of having one or more games to play after the main prize has been won, or of requiring the leader to play when there is nothing for him to win.

Handicapping.—This, so often necessary, calls for a nice knowledge of the contestants. Fixed rules are impossible. That one man has a chance to sit long and think while the other plays, and perhaps not always plays with as much ability as effect, makes billiards pre-eminently the temperamental game. There must, therefore, be much guessing in the name of handicapping. Not a few conductors of tournaments shirk their office by happily inveigling their players into handicapping themselves.

One thing is to be cautioned against. As a rule, if the light-weighted, with their imposts, about fairly balance the middleweights, they are apt to prove too heavy for the heavyweights. To illustrate, A can give B 30 in 100, B give C 30, C give D 20, and D give E 20. A in practice can possibly give C the 60 required by theory, but he can little better give D 80 than he can, as theory requires, give E 100 in 100! Again, if there are many entries, those with a light impost possess a decided advantage in having so much more to learn than the others. The oftener they play, the relatively better.

If A can give B 12 in 100, B give C 15, and C give D 23, then A should give C 25 and D 44, and B give D 35. It is all merely a question of multiplication, division, addition and subtraction, without being simple enough to look easy in print.

The process multiplies together the odds A gives B and B gives C, as 12×15=180, which is to be divided by the number of points (100) constituting game. The quotient, which is nearer 2 points than 1, is to be reckoned as 2, and deducted from the 15 B gives C, leaving 13, which, added to the 12 given B by A, makes 25 to be given by A to C. By a similar process—multiplying together the 15 given by B to C and the 23 given by C to D (15×23=345), dividing by 100 and subtracting the 3 from the 38 (15 added to 23)—35 are what B should give D. What A is to give D is ascertained by multiplying together the 15 (less 2) and the 23 (13×23=389), which, divided by 100, shows that 4 are to be deducted from the added 13 and 23, leaving 32, which, added to the 12 A gives B, makes 44 to be given by A to D.

Scoring Tournaments.—Owing to a faulty system of keeping track of games played, not a few conductors of tournaments are temporarily at a loss to determine with whom some contestants have yet to play. This formulary covers everything:

Figures next to names stand for handicap, if any.

Figures standing alone in squares are for total first, average next, and highest run last.

Winning and losing averages are both given, and in common fractions, with the double purpose of showing which player led (in case of later dispute), and of facilitating the making-up of general and tournament averages when play is done.

When a game is over, add an I to Games Won and an X to Games Lost.

When tournament is finished add up totals, as well as innings (last figures of those in middle line of squares), and compute single, general and tournament averages decimally.

To find out who has yet to play, look for blank spaces exclusive of those running obliquely and marked D. (for Jones of horizontal column doubling with Jones of vertical). In the table are four blanks, meaning two games to play—Gray with Jones and Smith.

To find out how many games have been played, add I’s and X’s together, and divide by 2.

When I’s and X’s differ in their totals, there has been an error in tallying games either won or lost.

Scoring for the Press.—Care should be taken to begin with the score of him who plays first. His winning then will mean that the innings were unequal, while putting the winner’s score second will indicate equal innings. Disregard of this rule, prevalent of late years, forces whoever would verify the average to count up the innings in each score.

Decimals are Best.—Divide total points by total innings. Thus, 300 points in 28 innings show 1020
28 in crude fractions, 10 and 5/7ths in the lowest evenly reduced ones, and 10.71 (71/100ths) decimally. The first system seldom gives an accurate idea at sight. In the second, the fractions cannot always be reduced evenly, as above. Ordinarily, the third is closest, briefest and clearest.

Avoid a Jumble.—Some computers mix themselves and others up by using all three methods. Others, as a convenience, express the single average as 1020
28, and the general average not as 8170
175, to be consistent, but as 8.97. This is akin to the barbarism of speaking in two languages at once. There are others who, simply because it is so divisible, convert the 8170
175 into 834
35, so that anybody seeking to prove the average by finding the points and innings will have rare figuring as a preliminary.

Decimalizing.—This is simply adding a cipher to the right-hand end of every remainder after the dividend has no unused figure left. Adding a cipher to the 20 in 1020
28 yields 7 and 4 over when divided by 28, and now adding a cipher to the 4 will result altogether in 10.71, with 12 over.

Pay no attention to this remainder unless, if a general average, 10.71 seems to be a tie with some other general average. Such a tie will rarely happen. Should it, add a cipher to the 12, and dividing the 120 by 28 will result in 10.714 (1000ths now, instead of 100ths), with 8 over. If there is still a tie, proceed as before, first making 80 of the 8.

Give and Take.—Had the 10.71’s remainder been 14 or more, instead of 12, which is less than one-half the innings, the average would change to 10.72. The arbitrary rule is to ignore the final remainder when it is less than half the innings, but enlarge it and give it to the player when it is half or more.

Reconversion.—If for any reason it be necessary to find the number of innings, add ciphers (two will usually be enough in billiards) to the points, and divide by the decimalized average. Thus 1071)30000(28 innings, with 12 over. To find the points on which a general average is based, innings (50) and average (16) being known, multiply the one by the other—16×50=800.

General Average.—A match of continuous points has but one average, whether it be played in one session or half a dozen; but it is different both in a tournament and in a match of several separate games, a majority to win.

In computing the general average, avoid the easy error of adding all a player’s game-averages together, and dividing the product by the number of games. There is only one condition in which this will show the true average, and that is when all the games have innings separately equal in number, howsoever much the points themselves may vary.

Illustration of false and true:

The average found by dividing by the number of games is grossly extravagant.

Losing Averages.—Properly, the loser’s average can never be higher than the winner’s. To concede that it can is to premiumize its maker’s inefficiency. Setting out to win the opening shot, he had failed, which is the only way, with fewer points, to make the seemingly higher average. It is equally unfair, in a continuous game of several sessions, to concede an average for a fraction of the game. By getting far behind, one player is without limit on any night, while the other is stopped every night by reaching the number of points assigned to every leader.

Except as personal compliments, losing averages are valueless. Their apparent makers do not wholly make them. Much depends upon the other man. The loser reaches a high figure largely because, having aimed to cover a given number of points, he failed to do so. It has often happened that a player with 50 to go has needed as many innings to make them as he had taken to make his other 250. As a rule, losers “let down” near the finish more than winners, and hence their average is dependent less upon themselves than upon those who close the game.