Non Abstarct Mathematic

Non abstract mathematics

Mathematics is abstract by nature, and mathematicians are bunch of weirdos who speak in tongues. At least, this is the picture that comes to most lay person, myself included. However, mathematics can be, and should be, explained in simple and comprehensible terms so that normal people like us who don’t understand the jargons and tongues can grasp the basic concepts. Real mathematicians don’t bother to explain the basic concepts to lay persons; it’s probably sort of trade secrets that they are reluctant to reveal, lest that might make them less special.

Hence, I will try to demonstrate that mathematics is part of our daily life and can be explained with simple diagrams. Of course, at more complex levels, mathematics is still an abstract subject that requires comprehensive understandings of the respective equations and algorithms. But, that’s not our concern, that’s the job of tongue speaking mathematicians.

First, let’s look at one very simple problem which we would face at least once in 4 years, if you are a soccer fan.

If there are 5 soccer teams in one group, how many matches will be played if every team has to play the other teams in the same group?

The most direct and simplest way to work this out is to write the teams on a piece of paper and start matching them to each other. Of course you would get the right answer. No magic about this. But is there a simpler, more elegant way to do this without evoking the mumbo jumbo of mathematics equations and what not?

Let’s reduce this to a simpler scenario. There are only two teams in a group. So, how many matches they have to play? Even a dumbo with IQ below 80 points like me can answer this question. One match! OK, now how about three teams in one group, or four teams in one group? It gets a bit complex as the number of teams grows in a group. To have a generic and yet simple way to look at this problem, simply draw the number of teams in the group as dots on a piece of paper and then join up all the dots, such that every dot is connected to all other dots.

For example, two teams in a group are just two dots and a line joining them. The total number of lines will be the total number of matches played in the group.

Two teams in a group – one match will be played.

3 teams in a group – 3 matches will be played.

4 teams in a group – 6 matches will be played.

5 teams in a group – 10 matches will be played.

Now, isn’t this a more elegant way to represent the problem and much simpler to remember? The best thing is, you don’t have to start matching the teams and count the matches every 4 years. Of course, it does get a bit complicated if the number of teams in a group is more than 6. But, there are only 32 teams in the world cup final and each group usually has less than 6 teams.

Now, let’s go one step further. What is we are not dealing with soccer teams but a card game that requires 3 players to play in each round. If there are 5 players in a group, and each game is played by 3 players, how many games will be played if each player has to play all other players in the same group?

You may again start matching the players with pencil and paper, three at a time, until they are all matched and count the total matches. Alternatively, we can use the same diagrams; but instead of counting lines, this time we count the number of triangles with all three corners touching the dots.



3 teams in a group – only 1 game will be played.

4 teams in a group – 4 distinctive triangles, hence 4 games will be played.

5 teams in a group – 10 distinctive triangles, hence 10 games will be played.

If you can't see the triangles in the last figure, the following figures will help you to visualise. There are 5 such yellow triangles (left figure) and 5 such green triangles (right figure).

If you are keen, you may extend the same method to find out how many mahjung games will be played in a group of 5 players, if each player has to play every other player. For those who uninitiated, mahjung is a game played by 4 players.

Hence, instead of counting triangles, we count the total number of rectangles with all 4 corners touching the dots. In fact, this method can be extended for 5-players, 6-players and so on. You may try it to verify.

The best thing about this is – I figured this out on my own. Rub it in!

In fact, what we have seen has formal mathematical equations and theories associated with it. It is termed as “permutations and combinations”. To find the number of combinations of 2 teams out of 5 teams, the equation is:

      5!     .

(5-2)! x 2!

5! = factorial of 5 = 5x4x3x2x1

If we were to check against the formal mathematical equation, the answer would be the same, i.e. 10 matches. Because 5! is 120, (5-2)! is 6, and 2! is 2; and 120/(6x2) = 10.

The general equation for calculating number of combinations of matching r number of teams from n number of teams is:

      n!      .

(n-r)! x r!

Well, I did say mathematicians speak tongues, didn’t I?