Unit 2 Notes:

Forecasting Single Elimination Tournament - Determining Number of Days
 

There are basically three variables that a sports programmer needs to know in order to schedule a single elimination tournament.  They are the number of entries in the tournament (n), the number of days needed to conduct the tournament, and the number of games needed to play each day of the tournament.  By knowing any of these two variables, the programmer can figure out the third.  The first step for the programmer, therefore, is to figure out which piece of information they need to figure out.

For forecasting days, the programmer will know the number of teams in the tournament and the number of games that are played each day of the tournament.  The programmer is trying to determine how many days the tournament will take, if teams are ONLY PLAYING ONE GAME PER DAY.

Here's how it works....

Lets say that the following information is applicable for our tournament:

N= 27 teams
Number of games that can be played each day = 5

Step 1:  Determine the number of first round games
You can determine this by subtracting the NEXT LOWEST POWER OF 2 from the number of teams in your tournament.  In this case, 27 - 16 = 11 first round games.

Step 2:  Establish the number of rounds for the tournament.
You can determine this by determining the number of times 2 must be multiplied to equal or exceed N.  For example:
2 x 2 x 2 x 2 x 2 =32.  We multiplied 2 FIVE times...therefore, we will have 5 rounds in our tournament.

Step 3:  After the number of rounds have been determined, the next step is to enter the number of first round games (Step 1) under round 1 and always enter 1 game in the last round.

Round  1      2      3      4      5
Games  11                           1

Step 4:  Once the number of first and last round games has been listed, list the number of games in each round starting with the next to last round (in this case, the 4th round) and work our way back to the first round.  The number of games per round is always a multiple of 2.

Round  1       2       3       4       5
Games  11     8       4       2       1 = 26 games

Always check your answer by using the formula of N-1 for the total number of games.  In our case, 27-1 = 26 games.

Step 5:  After the number of games per round has been established, we can determine how many days are required to play the number of games in each round.  We do this by determining the number of games that can be played on a round by round basis (no team plays more than once per day).  Start with the last round and work backwards.  For rounds which only take one day to complete, when the number of games available per day is greater than or equal to the number of games listed in the round, place a 1 in that column.

Round  1       2       3       4       5
Games  11     8       4       2       1
Days                       1       1       1

When the number of games is greater than the number available per day, STOP!

Step 6:  Once we have reached the point where the number of games needed per day per round is greater than the number of games available per day, we go back to the first round and determine the number of days required to play each round.  We subtract the difference in games from the NEXT round in order to eliminate wasted games early in the tournament.

Round  1       2        3        4        5
Games  11     8 (4)  4        2        1
Days     3       1        1       1        1 = 7 days

Notice it will take 3 days to play 11 games if we can play 5 games per day.  Given that information, we will have 4 days left over (3 x 5 =15 games....we only need to play 11 games in the first round, so 15-11 = 4 games left over).  We subtract 4 from the 8 games in the second round, leaving us with 4 games to play.  If we can play 5 games per day, it will only take us 1 day to play the 4 games in the second round.

The total number of days needed to complete the tournament is determined by adding the number of days for each round.