Q1.
Here is an example from Līlāvatī — see if you can find the solution! Of a herd of elephants, half and one-third of the half went into a cave, one-sixth and one-seventh of one-sixth were drinking water from a river. One-eighth and one-ninth of one-eighth were sporting in a pond full of lotuses. And the king of the elephants was leading three female elephants. Tell me, how many elephants were there in the herd?
Answer
The herd had 1008 elephants.
Step 1 — write each group as a fraction of the herd. Let the herd be x elephants.
In the cave: half and one-third of the half
= x/2 + (1/3 of x/2) = x/2 + x/6
At the river: one-sixth and one-seventh of one-sixth
= x/6 + (1/7 of x/6) = x/6 + x/42
In the pond: one-eighth and one-ninth of one-eighth
= x/8 + (1/9 of x/8) = x/8 + x/72
With the king elephant: 1 king + 3 females = 4
= x/2 + (1/3 of x/2) = x/2 + x/6
At the river: one-sixth and one-seventh of one-sixth
= x/6 + (1/7 of x/6) = x/6 + x/42
In the pond: one-eighth and one-ninth of one-eighth
= x/8 + (1/9 of x/8) = x/8 + x/72
With the king elephant: 1 king + 3 females = 4
Step 2 — add the fractions. The LCM of 2, 6, 6, 42, 8 and 72 is 504.
1/2 = 252/504 1/6 = 84/504 1/6 = 84/504
1/42 = 12/504 1/8 = 63/504 1/72 = 7/504
Total = (252 + 84 + 84 + 12 + 63 + 7)/504 = 502/504
1/42 = 12/504 1/8 = 63/504 1/72 = 7/504
Total = (252 + 84 + 84 + 12 + 63 + 7)/504 = 502/504
Step 3 — the 4 elephants left over are the rest of the herd.
x − 502x/504 = 4
2x/504 = 4
x = 4 × 504 ÷ 2 = 4 × 252
x = 1008
2x/504 = 4
x = 4 × 504 ÷ 2 = 4 × 252
x = 1008
Step 4 — check it, group by group.
| Group | Working | Elephants |
|---|---|---|
| In the cave | 1008/2 = 504, and 504/3 = 168 | 504 + 168 = 672 |
| At the river | 1008/6 = 168, and 168/7 = 24 | 168 + 24 = 192 |
| In the pond | 1008/8 = 126, and 126/9 = 14 | 126 + 14 = 140 |
| The king and his three | 1 + 3 | 4 |
| Total | 672 + 192 + 140 + 4 | 1008 ✓ |
Why 1008 and not some awkward number: every division in the puzzle must come out whole — you cannot have two-thirds of an elephant. So the herd must be divisible by 2, 3, 6, 7, 8 and 9 together, that is by 504. Bhāskarāchārya chose 1008 = 2 × 504, and 1008 is also an auspicious number in Indian tradition. That is the craft of the Līlāvatī: the arithmetic is set up so that the answer falls out clean.
Did you know? The chapter says Bhāskarāchārya “had a gift for poetry and used this talent to make his writings enjoyable to scholars and students alike”. This is why — the sum is a story about elephants, not a page of fractions. Fig. 4.7 shows another Līlāvatī problem, drawn in a 17th-century manuscript with a peacock on a pillar and a snake below.