history of mathematics - National STEM Centre

7776 Arabs
Activity 5.4
This problem is believed
by one Arab author to
have originated, like
chess, in India.
Tlefore you run yo
program, refer to the
hints section.
60
The chess-board and other problems
1 The man who invented chess was asked by his grateful ruler to demand
anything, including half the kingdom, as reward. He requested that he be given the
amount of grain which would correspond to the number of grains on a chess board,
arranged in such a way that there would be one grain in the first square, two in the
second, four in the third, and so on up to the 64 squares. The ruler first felt insulted
by what he thought was a paltry request, but on due reflection realised that there was
not sufficient grain in his whole kingdom to meet the request. The total number of
grains required was 18 446 744 073 709 551 615. How did the chess-inventor find
this number?
2 a An example of an indeterminate equation in two unknowns is 3x + 4y = 50,
which has a finite number of positive whole-number solutions for (x,y). For
example, (14,2) satisfies the equation, as does (10,5). Find all the other pairs.
b Why do you think such an equation is called 'indeterminate'?
3 The 'hundred fowls' problem, which involves indeterminate equations, is found
in a number of different cultures. The original source may have been India or China
but it was probably transmitted to the West via the Arabs. Here is an Indian version
which appeared around AD 850.
I
Pigeons are sold at the rate of 5 for 3 rupas, cranes at the rate of 7 for 5
rupas, swans at the rate of 9 for 7 rupas and peacocks at the rate of 3 for 9
rupas. How would you acquire exactly 100 birds for exactly wrupas?
The author gives four different sets of solutions. Write an algorithm to find the
solutions, turn it into a program and run it. How many solutions are there?
4 Interest in indeterminate equations in both China and India arose in the field of
astronomy where there was a need to determine the orbits of planets. The climax of
the Indian work in this area was the solution of the equations
ox 2 ±c = y 2 and in particular ax 2 + 1 = y 2
where solutions are sought for (x, y).
|'A version of this problem is ftrst attributed by the Arab mathematicians to
iArchimedes who posed a famous problem known as the cattle problem, ^
I which reduced to finding a solution y of x 2 = I + 4 729 494_y2 which is |
^divisible by 9314. You can read about the cattle problem in, for example,
| The history of mathematics: a reader, by Fauvel and Gray, page 214, but
sdon't try to solve it. These equations are now known as Pell equations,
Inamed by Euler by mistake after an Englishman, John Pell (1610-1685),
fwho didn't solve the equation! The study of Pell equations has interested!
[mathematicians for centuries. In 1889, A H Bell, a civil engineer from
fillinois and two friends computed the first 32 digits and the last 12 digits!
;; of the solution, which has 206 531 digits.
Brahmagupta (born in AD 598) solved the Pell equation 8.x 2 +1 = y 2 using the
following method, called the cyclic method. It is presented, without explanation,
rather like a recipe.