"By one more than the previous number."
35² → 3×4 = 12, append 25 → 1225. Works for any number ending in 5!
Calculating squares of numbers ending in 5, multiplying numbers whose first digits add up to 10 and last digits are 5.
When squaring a number ending in 5, multiply the digit(s) before 5 by the next consecutive number, then append 25.
This sutra is based on the algebraic identity: (10a + 5)² = 100a(a+1) + 25. The beauty is that you only need to multiply the tens digit by "one more than itself" and write 25 at the end. This extends to any number ending in 5 — for example 105² = 10×11 followed by 025 = 11025. The sutra also applies when multiplying two numbers whose tens digits add to 10 and units digits are both 5, e.g. 35 × 75 = 3×8|5×5 = 2625. In the Vedic system this is considered one of the most elegant and useful shortcuts for mental squaring.
This sutra originates from the ancient Indian text "Vedic Mathematics" reconstructed by Bharati Krishna Tirtha (1884–1960), the Shankaracharya of Puri, who claimed to have rediscovered these 16 sutras from the Atharva Veda's appendix (Parishishta). The concept of "one more than the previous" reflects a deep Indian mathematical tradition dating back to the Sulba Sutras (800–500 BCE), where geometric constructions required rapid mental squaring. Indian mathematicians like Aryabhata (476 CE) and Brahmagupta (598 CE) used similar shortcut principles in their astronomical calculations, where squaring numbers was essential for computing planetary positions.
The algebraic proof is rigorous: (10a+5)² = 100a² + 100a + 25 = 100a(a+1) + 25. This identity holds universally in any base-10 system. From a cognitive science perspective, this sutra reduces a multiplication problem to a simpler one — computing a(a+1) requires less working memory than full squaring. Research in mathematical cognition shows that "chunking" complex operations into simpler sub-operations (as this sutra does) significantly improves speed and accuracy. The pattern also connects to the theory of finite differences in numerical analysis, where consecutive products form predictable sequences.