Choosing the helper
m must make x + y·m divide exactly by the difference, and m² should be as close to N as possible. The new sides are (x·m + N·y) ÷ |k| and (x + y·m) ÷ |k|.
29 / ANTIQUITY TO THE RENAISSANCE
Bhaskara II (Bhaskaracharya)
Can 61 equal square panels and one extra tile make a perfect square? Bhaskara found the answer, and a method that gets there.
Explore the mathematics ↓
01 / THE CYCLIC METHOD
From a near miss to an exact square.
N equal square panels plus one tile make a square exactly when x² − N × y² = 1.
A near miss leaves a small difference. One cycle of Bhaskara’s method turns it into another near miss, and cycles continue until the difference is 1.
Number of panels:
=
=
THE IDEAEach cycle turns x² − Ny² = k into another true whole-number relation, with a helper m that makes the division come out whole. With m² close to N, the differences stay small while x and y grow.
The person behind the ideas
- Led an astronomical observatory at Ujjain.
- Wrote Lilavati on arithmetic and Bijaganita on algebra.
- Also studied astronomy and trigonometry.
More of the mathematics
Why 61 is famous
Its smallest answer is x = 1,766,319,049 and y = 226,153,980. Trying panel sizes one at a time would take over 226 million tries.