MATHS WEEK IS under way and, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!
Astronomical Advances from India
Maths belongs to humanity – it is one of the greatest, most creative triumphs of the human mind.
Although advanced maths can be difficult to understand, we can appreciate its applications and also the stories of its creation and discovery.
Mathematics thrived in India from the end of the second millennium BCE, with the great flowering of Indian Maths from 400 CE right up to 1200 CE.
The decimal system is first recorded in India and zero – as we use it in our number representation – is first found there.
Religion, astronomy and astrology were important driving forces for mathematical progress as has been the case in other ages and cultures.
One of the great mathematicians of India’s great age of mathematics was Bhaskaracharya (Bhaskara the Teacher) who was born in 1114 in Vijayapura, India and became head of the astronomical observatory at Ujjain, where he died in 1185.
It would be several centuries before European Maths caught up in numerous areas.
Over the years, maths books have been stripped of story, lyricism or the personality of author. By convention they became cold and functional. However, that was centuries away for Bhaskaracharya, who dedicated a book to his daughter Lilavati.
The book’s 277 verses present different aspects of arithmetic, measurement, geometric and arithmetic progressions. The reader can test his or her knowledge with many entertaining puzzles.
Here is one of them:
A king sets out on an expedition to seize his enemy’s elephants.
His enemy’s city is 80 yojanas away. His army only marches 2 yojanas the first day.
Say, intelligent calculator, with what increasing rate of daily march did he proceed, since he reached his foe’s city, a distance of 80 yojanas, in a week?
If you like you can substitute miles or kilometres for yojanas; it doesn’t matter. In fact, you don’t need to use any units at all, that is the power of maths.
The question is suggesting that the king’s army increased the distance that they marched by the same amount each day.
So, they could have marched 2 units in the first day, three in the second, four in the third, and so on. If that was the case the answer would be 1 yojana.
(Incidentally, a yojana could be between 6 and 13 km, it varied depending on era and locality.)
All puzzles are by Eoin Gill, the coordinator of Maths Week Ireland and director Calmast STEM Engagement Centre, Waterford Institute of Technology
Come back tomorrow for Thursday’s puzzle and the answer for today’s.
Tuesday’s puzzle: The Answer
Their ages were 45, 60, 75
The method:
This puzzle may be approached using trial and error, but it should be spotted that their ages have the relationship of a Pythagoras Triangle.
There are many such triangles.
The 3-4-5 triangle is the best-known Pythagoras triangle (3-squared + 4-squared = 5-squared). But, any multiple of this triple will also be a Pythagorean triple such as 30-squared + 40-squared = 50-squared.
It is often useful to consider the most simple case or extreme cases. The smallest Pythagorean triple is 3, 4, 5.
If they were 3, 4 and 5 years old (simple case) their ages would differ by a year. The ages
differ by 15 so we might expect something useful to happen if we multiply 3, 4 and 5 by 15 where we get 45, 60 and 75.
We can see that these ages satisfy the conditions for the three Pythagoreans.
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