Single elimination tournaments are one of the most well-known types of tournament formats that sport programmers use.  There
are many advantages to programming a single elimination tournament, but like round robin tournaments, single elimination
tournaments also have some disadvantages.  Participants generally understand single elimination tournaments, and they produce
a true winner at the end of tournament play.

Single elimination tournaments also work well as post-season playoff formats after round robin tournaments are played.
Programmers are able to seed teams/players based on their performance in the round robin, allowing for better competition as
the tournament progresses.  If the highest seeds all "take care of business" and win their games, the #1 and #2 seeded teams
will face each other in the final.  Something a tournament programmer wants to avoid, however, is having the #1 and #2 seeds
face each other in a round OTHER THAN the finals.  Sometimes, it is not always apparent to the programmer how teams
should be seeded.  The programmer might not have insight regarding the teams' ability, or past performance.  In these cases, it
might be necessary to do a random draw to determine seeds, or the programmer may choose to seed based on order of entry.

We will see later that single elimination tournaments are based on powers of 2.  That is, when the tournament size is a power of
2 (2, 4, 8, 16, 32, 64, 128, 256, ...) there will be NO first round byes.  A "bye" game in a single elimination tournament only
occurs in the first round, and byes are generally awarded to the highest seeded teams.  Teams who receive a first round "bye"
will play their first game in the second round.  Byes are determined by taking the number of teams in the tournament (n) and
subtracting the number from the NEXT HIGHEST power of 2.  For example, in a single elimination tournament with 14 teams,
you would take the next highest power of 2 (16) and subtract 14 from it:

16-14 = 2

Therefore, there will be 2 byes in the first round for your single elimination tournament with 14 teams.

When drawing tournament brackets, it is important to FIRST have all of the teams or participants registered in your
tournament.  It is difficult to place teams in the tournament draw after you have already scheduled your tournament, so you
should make sure that all team entries have been received before scheduling your tournament.  Remember, single elimination
tournaments are based on powers of 2, so your tournament bracket should be balanced in such a way that from the second
round through the final round, the same number of teams will be in the upper half of the bracket as will be in the lower half of
the bracket.  This may not be true for the first round, due to the possible presence of bye games in the first round.

When you are ready to draw your tournament bracket, you may choose to draw and "implied bye" bracket or an "explicit bye"
bracket.  Implied bye games are not indicated on the tournament bracket....rather, blank spaces replace first round bye games
and only second round games appear.  Explicit bye brackets show all bye games by placing the word "bye" or drawing an "x"
through the bye game.  For the purposes of this course, it is easier to draw explicit bye brackets, to more easily accommodate
seeding and placing of bye games.  We will focus on this type of bracket from here on out.

Begin by taking the number of teams you have in your tournament....lets say 7.  Always use the bracket format for the NEXT
HIGHEST power of two...in our case here, you would use an 8 team bracket.  If you had 12 teams in your tournament, you
would use a 16 team bracket, if you had 23 teams in your tournament, you would use a 32 team bracket, etc.  You can see
examples of different sized brackets on pages 143-145 of the text.

After you choose your bracket, you must determine your tournament SEEDS.  Remember, the easiest way to determine seeds
is to look at past performance or team's ability.  If you do not know this information, random seeding is possible.  You can see
an example of how to seed a single elimination tournament bracket by referring to page 111 in the text.

After drawing the bracket and determining your seeds, you must place your BYE games.  The number of BYE games is equal
to the number of teams in your tournament subtracted from the next highest power of 2 (or in this case, the size of the bracket
you initially drew).  In our example, we have 7 teams in our tournament.  The next highest power of 2 is 8, so 8-7 =1, giving us
1 bye game.  Go down your tournament bracket and draw a line through every game that is represented by a seed higher than
7.  In our example, you would draw a line through the game matchup 1 vs. 8, because we only have 7 teams in the tournament.

Once you have completed placing your bye games, you are ready to schedule your tournament games!

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.
 

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.

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 determining the number of games needed to play per day, the programmer must know the number of teams in the
tournament and the number of days that it will take to play the tournament.  This occurs in situations where the programmer has
a knowledge of the number of teams he/she has registered, but is under a restriction of getting the tournament played in a
certain number of days.  The programmer then must be able to communicate to a facility coordinator that a certain number of
games MUST be played per day in order to get the tournament finished in the allotted number of days.

Here's how it works....

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

N= 27
Number of days that are available = 7

The first few steps are just like forecasting the number of days:

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.

OK, here's where we begin to differ from forecasting days....

Step 5:  After the number of games per round has been established, determine the number of days that must be available.  First,
divide the number of games in the tournament (26) by the number of days we have available (7).  This results in the FIRST
INDEX NUMBER, of (3.7).  ALWAYS ROUND THIS NUMBER UP!  Even if the Index Number was 3.1, you would still
round it up to 4.  Now, start with the last round, and work backwards.  For rounds whose number of games is less than our
first Index Number, place a 1 in that column.

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

Once we have reached the point where the number of games needed per day per round is greater than our first Index Number,
we must STOP!

Step 6:  Now, we must get a SECOND INDEX NUMBER.  Divide the remaining number of games in the tournament (19) by
the remaining number of days available (4, because we've already accounted for 3 of our 7 games available).  This results in our
SECOND INDEX NUMBER of (4.75).  Remember, always round up (5).  Working backwards from the last round
completed, we recognize the rounds whose number of games is less than the second index number.

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

Notice that in round 2, the number of games to be played (8) is still larger than the our index number (5).  We cannot go any
further.  When the index number is smaller than or equal to the number of games in the round, that index number becomes the
answer to the problem....we need 5 games available to play each day in order to get our 27 team tournament completed in 7
days.

It may be necessary, in some problems, to get a third index number or even a fourth index number before we cannot go any
further....continue to follow the steps above until you cannot progress further.  At that point, the LAST index number you figure
will be the answer to your problem.

You can double check the answer by using the forecasting method:

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

If you go back to the first round, you will realize that it will take 3 days to play 11 games, if you can play 5 games per day.
Account for the 4 games left over in the second round, and you will realize that it will take 1 day to play 4 games, if you can
play 5 games per day.  If you add up all of the days, you should come up with 7 total days....this is your check to see if your
index number is correct!

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 determining number of entries, the programmer will know how many games are available per day and how many days
he/she has to play the tournament.  This is a fairly common scenario...most of the time, we receive a facility reservation, and we
must determine how many teams we can accommodate in our tournament.  In this case, we are assuming that teams will not
play more than one game per day....

Here's how it works....

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

Number of games that can be played each day = 5
Number of days that are available = 7

Step 1:  The first step in solving the problem is to determine the number of total possible games that can be scheduled in a single
elimination tournament.  Do this by multiplying the number of days that are available (7) by the number of games that can be
played each day (5).  The answer will be the TOTAL POSSIBLE number of games that can played in the tournament (35).

Remember, in single elimination tournaments, you will never use all of the possible games available to you...you will always end
up wasting games.  For example, if we have 5 games available to play each day, during our final round, we will only use 1 of
these games.  That means in the final round, we will be wasting 4 games.  We must figure out how many TOTAL games we will
waste throughout the tournament, and subtract that number from the TOTAL POSSIBLE number of games that can be played
(35).

Step 2:  Begin with the last round of the tournament.....how many games will always be played in the last round?  Thats right,
the answer is 1.  We know that if we can play 5 games in a day, and we only are playing 1 game in the last round, we will be
wasting 4 games.

How about the other rounds?  Work backwards from the last round and determine the number of games that will be
played....remember, we use powers of 2 to figure this out.  Continue to figure out the number of wasted games by subtracting
the number of games in the round from the number of games that can be played each day (5).  In the next to last round, you will
play 2 games.  If you can play 5 games in one day, you will be wasting 3 games.  Continue working backwards until you hit a
point where you will not be wasting any games....in this case, when you hit 8 games in a round, you won't be wasting any of the
games you will play in one day.  You will use all 5 games.  At this point, you STOP, and add the number of wasted games.

Round    R        R        R        R        Last
Games    16      8         4        2             1
Wasted    0       0         1        3             4 = 8 wasted games

Step 3:  Subtract the total wasted games (8) from the total number of possible games (35).  Your answer will be 27.  This
represents the ACTUAL number of games that can be played in your tournament.

Step 4:  The total number of entries that can be accommodated in the tournament is determined using the following formula:

Number of games = N - 1

27 = N - 1
28 = N