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If you have ever groaned at a word problem, you have something in common with students eight centuries ago. Bhaskara II, known in India as Bhaskaracharya, wrote his arithmetic book, the Lilavati, almost entirely as word problems about kings, peacocks, lotus flowers and monkeys. This post works through four of them with Class 10 tools, looks at where he went wrong about zero, and turns his method into a practice routine.
Who was Bhaskara II?
The MacTutor biography from the University of St Andrews gives his dates as 1114 to 1185 and says he became head of the astronomical observatory at Ujjain, the leading mathematical centre in India at the time, where Brahmagupta had worked before him. Henry Colebrooke's 1817 translation, Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, records Bhaskara's own statement that he was born in 1036 of the Saka era, which is 1114 CE, and that he completed his astronomical work, the Siddhantasiromani, at 36. Colebrooke concludes that the Lilavati dates to 1150 at the latest.
MacTutor lists six known works. The Lilavati covers arithmetic, progressions, geometry, the kuttaka method for whole-number equations, and combinations. The Bijaganita covers algebra. The Siddhantasiromani covers mathematical astronomy and the sphere.
You may have heard that he wrote the Lilavati to console his daughter after a failed wedding prediction. MacTutor calls this "a charming story" but says it is hard to find any evidence for it, and that it is not even certain Lilavati was his daughter.
Problem 1: how fast did the king march?
From the Lilavati, in Colebrooke's translation: "On an expedition to seize his enemy's elephants, a king marched two 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 eighty yojanas, in a week?"
This is an arithmetic progression with first term a = 2, number of terms n = 7 and sum S = 80. Find the common difference d.
Use S = (n/2)[2a + (n − 1)d], so 80 = (7/2)[4 + 6d].
Multiply both sides by 2/7: 4 + 6d = 160/7.
Subtract 4, which is 28/7: 6d = 132/7.
Divide by 6: d = 22/7 yojanas a day.
Colebrooke's text gives the same answer. Bhaskara's own rule, divide the sum by the number of days, subtract the first term, then divide by half of one less than the number of days, gives (80/7 − 2) ÷ 3 = (66/7) ÷ 3 = 22/7. Our guide to arithmetic progressions in Class 10 has more like this.
Problem 2: where did the peacock catch the snake?
"A snake's hole is at the foot of a pillar, and a peacock is perched on its summit. Seeing a snake, at the distance of thrice the pillar, gliding towards his hole, he pounces obliquely upon him. Say quickly at how many cubits from the snake's hole do they meet, both proceeding an equal distance?" Colebrooke's statement takes the pillar as 9 cubits, so the snake starts 27 cubits away.
Let the meeting point be x cubits from the hole. The snake travels 27 − x. The peacock flies the hypotenuse of a right triangle with sides 9 and x, and the two distances are equal.
9² + x² = (27 − x)²
81 + x² = 729 − 54x + x²
54x = 648, so x = 12 cubits.
Each travels 27 − 12 = 15 cubits, and 9² + 12² = 81 + 144 = 225 = 15², so the answer checks.
The same chapter has a bamboo, 32 cubits tall, snapped by the wind so that its tip touches the ground 16 cubits from the root. It broke 12 cubits up: the upright part is 12, the fallen part is 20, and 12² + 16² = 400 = 20².
Problem 3: how many monkeys?
This one is from the Bijaganita: "The eighth part of a troop of monkeys, squared, was skipping in a grove and delighted with their sport. Twelve remaining were seen on the hill, amused with chattering to each other. How many were they in all?"
With x monkeys, (x/8)² + 12 = x. Multiply by 64: x² − 64x + 768 = 0. Then x = [64 ± √(4096 − 3072)]/2 = (64 ± 32)/2, so x = 48 or x = 16. Colebrooke's text gives both values, 48 and 16, and both work: 6² + 12 = 48 and 2² + 12 = 16. MacTutor notes that Bhaskara treated the two solutions as equally admissible. See our guide to quadratic equations in Class 10.
Where did Bhaskara go wrong?
Bhaskara improved on Brahmagupta's rules for zero, but division still defeated him. In the Bijaganita, as quoted by MacTutor, he says that a quantity divided by zero "is termed an infinite quantity". MacTutor explains the problem: if n/0 were a number called infinity, then 0 × infinity would have to equal every n, so all numbers would be equal. The safe statement today is that division by zero is not defined.
The Lilavati even contains a puzzle built on this idea. A number is multiplied by zero, half of the result is added, the total is multiplied by 3 and then divided by zero, and the answer is 63. Treating the multiplication and division by zero as cancelling, (x + x/2) × 3 = 63, so Colebrooke's text gives x = 14. The idea is that multiplying by zero and then dividing by zero restores the number. In modern terms, that is closer to cancelling a factor that tends to zero in a limit than to arithmetic with zero itself.
What can a student take from Bhaskara?
Practise with stories, not just symbols. Exams set problems in words. Bhaskara's book is a reminder that translating words into an equation is a skill to drill, separately from solving.
Check every answer in the original story. 12 cubits works because both creatures travel 15. A root that fails the story, such as a negative length, should be rejected; one that passes, like the second troop of monkeys, should be kept.
Keep a rule for the cases that break. His one weak spot was zero. Know where your own methods stop working.
When you can solve the equation but cannot see how to set it up from the words, the Fermi Problem Library has a step-by-step solver that explains the reasoning, including the set-up line, rather than printing only the answer.
A short practice routine
Solve the lotus problem from the same chapter: a lotus bud stands half a cubit above the water and, pushed by the wind, sinks at a point 2 cubits away. How deep is the water? (Answer: 15/4 cubits.)
For three word problems this week, write the unknown, the figure and the "equal" sentence before any algebra.
Pick one problem from this post, close the page, and solve it again tomorrow from the words alone.
