The story behind a dice game

COMBINATORICS

0. Combinatorics – The story behind a dice game

Every day we come across passwords. Passwords for computers, passwords for ATMs (commonly referred to as pin number), number locks, and so on. All of these are combinations; some are combinations of alphabets while others are combinations of numbers and still others use symbols; in some cases the passwords are case-sensitive, sometimes they are not.

Among other examples of combinations we can think of – how your teachers come up with the time table, how a metallurgist tries different combinations of elements to come with up an alloy of desired properties, how a linguist examines the meanings of combinations of letters in an unknown language and so on. All of these visibly 'different' applications come under one roof called combinatorics. Combinatorics (or combinatorial mathematics) is a field of mathematics that deals with problems of how many different combinations can be built out of a specific number of objects.

This field has its origin in the gambling games that played a large part in the European high societies in the 16th century. Whole fortunes were won or lost in a game of cards or dice; something very similar to how the Pandavas lost all their fortunes in a game of dice in the Mahabharatha! In how many ways can a certain sum in throws of two or three dice be scored (haven't you played Ludo?), in how many ways is it possible to get two kings in a card game and other similar problems in a game of chance gave the initial push to develop combinatorial mathematics and the theory of probability.

Italian mathematician Tartaglia was among the first to list the various combinations that can be achieved in a game of dice. His list showed the number of ways 'x' dice can fall. However he failed to take into account the fact that the same sum can be achieved in different ways. For example, if we are using 2 dice and we want a sum of 7, the various combinations are (1,6), (2,5) and (3,4).

In the 17th century, Chevalier de Mere, an ardent gambler, had sort the help of his friend Pascal to determine the division of the stakes of an interrupted game of chance. This marked the first theoretical investigation into the problems of combinatorics. Fermat, a contemporary French mathematician, was also working on the same problem. Their work was followed by valuable contributions from Bernoulli, Leibnitz and Euler.

Combinatorics is extensively used in the field of statistics, cryptography, discrete mathematics, linear programming, group theory, non-associative algebra... the list is unending. Most of the names given above might sound new and not of your understandability. However, it is interesting to realise that the mathematics involved in all of them is the same as in a game of dice. Through a series of articles we will travel with Kabani (a student like you) through the field of combinatorics. We will learn to solve problems from the simplest to the toughest, and enjoy the beauty of mathematics. The only thing that you need to know is how to play a game of dice!

Food for thought -

Simplest Question – In a class, every student is to be given a 2-digit roll number. What is the maximum number of students that can be given the roll number?

Toughest Question – There is a queue of x + y persons at a ticket counter of a cinema theatre. x have Rs20 note and y have Rs10 note. Each ticket costs Rs10 and the cashier has no change to start with. In how many ways can the people line up so that the line keeps moving and no one has to wait for change?

Get the dice rolling... try solving the above problems. Correct solutions will get prizes.

Reference – “Combinatorial Mathematics for Recreation” by N. Vilenkin. Translated from the Russian by George Yankovsky