If you know you maths, the nightmare should begin if you are cheering for Brasil

THE PROJECTED SCENARIOS UNDER WHICH BRAZIL CAN CRASH OUT OF THE 1ST ROUND -  BB SUBBA IJAM
The first round, or the group stage, of the format in which the FIFA World Cup tournament is being played, throws up interesting mathematical possibilities presenting a delight for a mathematics-minded football fan to analyse and predict the scenario for his favourite team especially after all the 4 teams of the group that his favourite team belongs to have played two games each and the 3rd and last game of each team (which translate into last two matches) in the group are awaited.
In applying mathematical tools to solve the ‘problem of scenario’, one has to understand the relevant FIFA rules (see box). Only two top teams ranked on the basis of these rules qualify for the second or the knockout round. The top ranking teams of group-A are then identified as A1 and A2, that of group B as B1 and B2 and so on. Further, finishing as either A1 or A2 for a team will make a huge difference in the sense that they will encounter entirely different set of opponents in the knockout round.  
It is possible to mentally solve the entire scenario for a particular group by imagining the different possible outcomes of the last two matches. However, applying a little bit of mathematics will give more correct and accurate solutions.
To proceed then, there is more than one way to convert the FIFA group-stage rules given in the box into a mathematical model consisting of algebraic and logical expressions. These models can be easily implemented with the help of Microsoft Excel. To keep it simple, one can implement only that part of the model that deals with the first three criteria (let’s call it criteria abc, please see the box for details) and consider the more advanced criteria (let’s call it criteria def, please see the box for details) only after criteria abc places two teams at equal ranking. Thereafter, one can simply simulate the entire scenario for the last two matches of a group by entering imaginary goals to create a win, a draw and a loss for each team and even the goal difference wherever relevant.
The last two matches of group A in the ongoing World Cup are Brazil vs Cameroon and Mexico vs Croatia, both being played out on 23 June 2014. The FIFA rule is that both the last matches of a group - which always involve all the teams at once - are played at exactly the same time to ensure that result of one match does not unfairly affect the performance of teams in the other match. Brazil, the present host country, is among the teams commanding highest fan following across the world irrespective of caste, creed or nationality. All fans of Brazil are obviously anxious to know under what scenario their favourite team will progress or crash out of the tournament. Each team, including Brazil of group A has played two of the required three matches and their current standing is given in the accompanying table.
As is seen in the table, although Brazil and Mexico both have 4 points each, the former is placed at the top because of its better goal difference (GD) over Mexico.
Since there are three possible outcomes for a team in a match - a win, a draw or a loss - applying the rule of combination there will be 3x3= 9 possible scenarios based on the possible outcome of the last two matches of group A, without taking into account the factors GD, GF, GA that is. For applying the criteria ‘abc’, there is one important but very simple axiom which will greatly simplify the analysis. The axiom can be stated as under:
“Law of effect of results on goal difference: In the future match of a team, a win will increase, a loss will decrease and a draw will have no effect on the standing goal difference of a team.”
So, applying the above law with established mathematical theory of combination, the following possible scenarios for group A emerge (remember that Brazil is playing Cameroon and Mexico with Croatia):
Conclusions: From the above possible scenario, the following interesting conclusions can be drawn:
1. Out of the nine possible scenarios, assuming half a chance for both Brazil and Mexico to finish as A2 in scenario 9, overall Brazil has roughly about 7.5/9 (83%), Mexico 6.5/9 (72%) and Croatia 4 (44%) chances of going to the second round;
2. Cameroon cannot qualify for the 2nd round; they are out now and their last match with Brazil is a mere obligation as far as they are concerned. But they can still play spoilsport for Brazil if they manage to win as explained below;
A weird scenario: Since both Brazil and Mexico have 4 points each at present and are separated only by goal difference, Brazil’s fans would like to think that a loss for Brazil and a win for Mexico would be the worst scenario for Brazil which is not true as can be seen from the above table; Brazil will still qualify for the 2nd round as A2 in this scenario. One of the two scenarios in which Brazil will fail to qualify for the second round will be- hold on your breath if you too are a Brazil fan- when they will lose to Cameroon and Mexico play a draw with Croatia! In such a scenario, Mexico will get A1 position with 4+1=5 points whereas Croatia (3+1) and Brazil (4+0) will finish with 4 points each. Since both Brazil and Croatia have the same value of goal difference at ‘2’ before the their last matches (see the team standings table), a draw for Croatia will not change its present goal difference of 2 (recall the law stated earlier) whereas a loss for Brazil will reduce its goal difference from present 2. Thus Croatia will qualify as A2 from this group thereby eclipsing the entire universe of Brazil fans!
3. Scenario in which goal difference will make the real difference: Consider the scenario 9: Brazil lose, Mexico lose. In this case, Croatia will top the group with 3+3=6 points whereas Brazil and Mexico will have to compete with each other for A2 position with their present points of 4 each. Since both the teams will have lost to their respective opponents in this scenario, their goal difference will also reduce. Commonsense says that the team finishing with lower overall goal difference will lose out. True, but by exactly what goal difference? Again, applying a little mathematics on the table of current standing will make it clear that for Brazil to remain in the tournament, the difference of goal difference of Brazil and Mexico in their respective last matches must not be more than 1. In other words, if Mexico loses to Croatia by 1 goal then Brazil should not lose to Cameroon by more than 2 goals. Similarly if Mexico loses by 2 goals then Brazil should not lose by more than 3 goals and so on.
Yes, it’s complicated, so if you are a Brazil fan, pray that they keep it simple and beat Cameroon convincingly to scratch off the possibility of any nightmare.
[the writer can be contacted at bibasu1728@gmail.com]

Here is the rule for the ranking of the teams in the Group Stage of the World Cup, from Article 41 of the Regulations, 2014 FIFA World Cup Brazil, courtesy http://www.conmebol.com/en/ content /tie-breaker-rules-group-stage-2014-world-cup-brazil.

COMPETITION

The league format shall be used: each team playing one match against each of the other teams in the same group, with three points for a win, one point for a draw and none for a defeat.

The ranking of each team in each group shall be determined as follows:

 a) Highest points obtained in all group matches;

 b) Goal difference in all group matches;

 c) Highest number of goals scored in all group matches.

If two or more teams are equal on the basis of the above three criteria, their rankings shall be determined as follows:

 d) highest number of points obtained in the group matches between the teams concerned;

 e) goal difference resulting from the group matches between the teams concerned;

 f) greater number of goals scored in all group matches between the teams concerned;

 g) drawing of lots by the FIFA Organizing Committee.

 The teams that qualify from the group stage will play the round of sixteen.

Sl No	POSSIBLE OUTCOME	WHICH TEAM WILL GO TO 2ND ROUND		REMARKS
		A1	A2
1	Brazil win, Mexico win	Brazil	Mexico	A1, A2 by  points and goal difference
2	Brazil win, Mexico draw	Brazil	Mexico	A1, A2 by points alone.
3	Brazil win, Mexico lose	Brazil	Croatia	A1, A2 by points alone.
4	Brazil draw, Mexico win	Mexico	Brazil	A1, A2 by points alone.
5	Brazil draw, Mexico draw	Brazil	Mexico	A1, A2 by  points and goal difference
6	Brazil draw, Mexico lose	Croatia	Brazil	A1, A2 by points alone.
7	Brazil lose, Mexico win	Mexico	Brazil	A1, A2 by points alone.
8	Brazil lose, Mexico draw	Mexico	Croatia	A1 by points and A2 by points and goal difference
9	Brazil lose, Mexico lose	Croatia	Brazil or Mexico	A1 by points and A2 by goal difference of Brazil and Mexico

(Abbreviations used: GP- games played, W-won, D-draw, L-lost, GF-goal for, GA-goal against, GD-goal difference, PTS- points)
#	Country	GP	W	D	L	GF	GA	GD	PTS
1	Brazil	2	1	1	0	3	1	2	4
2	Mexico	2	1	1	0	1	0	1	4
3	Croatia	2	1	0	1	5	3	2	3
4	Cameroon	2	0	0	2	0	5	-5	0