Practical Application of Laws of Indices: 2018 FIFA WORLD CUP RUSSIA Fixtures
If you have not already seen my article explaining the Laws of Indices as algebraic techniques, feel free to check it out now. This article below explains how the Indices Laws can be applied to designing FIFA World Cup Fixtures, using the case of Russia 2018. There is also a brief user challenge at the end to get you thinking about how you may apply Indices Laws yourself.
The World Cup starts off with about 32 countries. These 32 teams/countries are vetted through 6 levels of competition, where Algebraic techniques like Indices is mostly applied to calculate the remaining number of teams per stage.
To give you time for a second coffee cup, you may click here to access how Laws of Indices have been effectively applied to designing FIFA World Cup Fixtures. The case of RUSSIA 2018 World Cup has been used as an example.
Below is a description of various competitive levels and the application of Indices Laws in calculating total number of remaining teams.
GROUP STAGE [32 Countries]
This is made of 32 countries put into 8 groups with alphabetical labels A to H.
32 / 2² = 2⁵ / 2² = 2⁵ × 2^(-2)
Adding the two exponents
=> 2^(5–2) = 2³ = 8
Each of the 8 groups will contain 4 teams calculated as; 4×8 = 32 countries.
GROUP A
In Group A, teams will be matched with themselves (Each team will play against three other involved teams) as described in the Group A matches diagram below;
Group A Matches
At the end of this group match, the 1st and 2nd winners of Group A will be put into one group and be considered qualified for the Round of 16 stage. Since only two(2) teams, placed 1st and 2nd positions can qualify for the next stage, we can calculate these two teams with Laws of Indices as below;
Calculation for Teams with Indices Law 2
Let number of teams in Group A= GA = 4
Let number of teams who could NOT qualified for Round of 16 = ALost= 2
Let number of teams who qualified for Round of 16 = AWin= unknown
Expressing the variables as indices;
GA = 4 =2²
ALost = 2 = 2¹
Since the number of teams who do not qualify will be taken off the total number of teams to get the number of teams who qualify;
AWin = GA × (1/ALost) =2²/(2¹) = 2²× 2^(-1)
Adding the powers using the 2nd Indices Law;
AWin = 2^[2 +(-1)] =2^¹= 2
Thus, the number of qualified teams from Group A, AWin for the Round of 16 is 2.
This calculation will be same for number of teams that qualify for the Round of 16 from the seven (7) remaining groups. (Group B to H).
GROUP B
In Group B, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group B matches diagram below;
Group B Matches
At the end of this group match, the 1st and 2nd winners of Group B will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
GROUP C
In Group B, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group B matches diagram below;
Group C Matches
At the end of this group match, the 1st and 2nd winners of Group C will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
GROUP D
In Group D, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group D matches diagram below;
Group D Matches
At the end of this group match, the 1st and 2nd winners of Group D will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
GROUP E
In Group E, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group E matches diagram below;
Group E Matches
At the end of this group match, the 1st and 2nd winners of Group E will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
GROUP F
In Group F, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group F matches diagram below;
Group F Matches
At the end of this group match, the 1st and 2nd winners of Group F will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
GROUP G
In Group G, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group G matches diagram below;
Group G Matches
At the end of this group match, the 1st and 2nd winners of Group G will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
GROUP H
In Group H, teams will again be matched with themselves (Each team plays against three other involved teams) as described in the Group H matches diagram below;
Group H Matches
At the end of this group match, the 1st and 2nd winners of Group H will be put into one group and be considered qualifiers for the Round of 16 stage. Only two(2) teams, placed 1st and 2nd positions can qualify for the next stage. To calculate for these two qualified teams, refer to example in Group A (Calculation for Teams with Indices Law 2).
ROUND OF 16 [16 Countries]
This is the second stage of the competition made of 8 pairs of matching teams(16 Countries). These eight pairs are mainly the first and second teams who emerged qualifiers for Round of 16 stage from the previous groups A to H. Below is a demonstration of the matches for Round of 16. Let first number represent position of team, and second letter represent Group name of team. For instance,1A will be the name for the team which came 1st from Group A, AWin and we can do same for the other groups as below;
1A — Winner from Team A
2A — 2nd Winner from Team A
1B — Winner from Team B
2B — 2nd Winner from Team B
1C — Winner from Team C
2C — 2nd Winner from Team C
1D — Winner from Team D
2D — 2nd Winner from Team D
1E — Winner from Team E
2E — 2nd Winner from Team E
1F — Winner from Team F
2F — 2nd Winner from Team F
1G — Winner from Team G
2G — 2nd Winner from Team G
1H — Winner from Team H
2H — 2nd Winner from Team H
Match 01
This will be a football match between teams 1C and 2D. The winner of this match will emerge as QF1, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 02
This will be a football match between teams 1A and 2B. The winner of this match will emerge as QF2, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 03
This will be a football match between teams 1B and 2A. The winner of this match will emerge as QF3, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 04
This will be a football match between teams 1D and 2C. The winner of this match will emerge as QF4, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 05
This will be a football match between teams 1E and 2F. The winner of this match will emerge as QF5, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 06
This will be a football match between teams 1G and 2H. The winner of this match will emerge as QF6, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 07
This will be a football match between teams who qualified QF1 and QF2. The winner of this match will emerge as SF1, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
Match 08
This will be a football match between teams 1H and 2G. The winner of this match will emerge as QF8, and will compete in the Quarter Finals. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the knockout phase for 2018 FIFA World Cup Russia.
QUARTER FINALS [8 Countries]
Quarter Finals 01
This will be a football match between teams that emerged as QF1 and QF2. The winner of this match will emerge as SF1, and will compete in the Semi Finals of the FIFA World Cup. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the Quarter Finals phase for 2018 FIFA World Cup Russia.
Quarter Finals 02
This will be a football match between teams that emerged as QF5 and QF6. The winner of this match will emerge as SF2, and will compete in the Semi Finals of the FIFA World Cup. The team which comes second will then be sent home and be out of the competition. Below is a sample case of the Quarter Finals phase for 2018 FIFA World Cup Russia.
Quarter Finals 03
This will be a football match between teams that emerged as QF7 and QF8. The winner of this match will emerge as SF3, and will compete in the Semi Finals of the FIFA World Cup. The team which comes second is sent home and taken out of the competition. Below is a sample case of the Quarter Finals phase for 2018 FIFA World Cup Russia.
Quarter Finals 04
This will be a football match between teams that emerged as QF3 and QF4. The winner of this match will emerge as SF4, and will compete in the Semi Finals of the FIFA World Cup. The team which comes second is sent home and taken out of the competition. Below is a sample case of the Quarter Finals phase for 2018 FIFA World Cup Russia.
SEMI FINALS [4 Countries]
There are two rounds of the Semi Finals where each group (made of a 2 winners from the Quarter Finals(SF1, SF2, SF3 and SF4) compete in a match. Out of the Semi Finals will emerge 1st, 2nd and 3rd. In the first match, teams that qualified for SF1 and SF2 will be matched in a competition. In the second match, teams that qualified for SF3 and SF4 during the Semi Finals will be matched in another competition. Below is a sample case of the Semi Finals phase for 2018 FIFA World Cup Russia.
Semi Finals 01
This will be a football match between teams that emerged as SF1 and SF2. The winner of this match will emerge qualified for the FIFA World Cup Finals, while the other losing team is qualified for play off third place. Below is a sample case of the Semi Finals 01 phase for 2018 FIFA World Cup Russia.
Semi Finals 02
The match is between teams that emerged SF3 and SF4. The winner of this match will emerge qualified for the FIFA World Cup Finals, while the 2nd team is qualified for play off third place. Below is a sample case of the Semi Finals 01 phase for 2018 FIFA World Cup Russia.
PLAY OFF THIRD PLACE [2 Countries]
Teams that emerged 2nd (second) in each of the two rounds for Semi finals will compete in the Play Off Third Place match. The winner in this match is placed 3rd in the FIFA World Cup Finals.
FIFA WORLD CUP FINALS [2 Countries]
Teams that emerged 1st (first) in each of the two rounds for Semi finals will compete in the Finals match. The winner in this match is placed 1st in the FIFA World Cup. The second winner is also placed 2nd in the FIFA World Cup Finals.
The above illustration tells how applicable Algebra and/or Mathematics is to everyday life. I hope to see your feedback or comments on this read.
Simple Assignment for Readers :)
Can you apply any of the Indices Laws in finding which months you consume the highest bandwidth of internet ?
I am happy to hear your feedback in the comments section. You may always roll back to my previous article explaining the Laws of Indices.
