Table of contents

  1. Research: I think I just found a new way to do large multiplications easily
  2. The Idea
  3. Example 1: 996 × 1007
  4. Example 2: 876 × 943
  5. How I found this and some thoughts
Research: I think I just found a new way to do large multiplications easily Research: I think I just found a new way to do large multiplications easily

Research: I think I just found a new way to do large multiplications easily

8/19/2025 / 4 minutes to read / Tags: math, discovery, research

The Idea

Suppose you want to compute a × b.

  1. Compute the sum: S = a + b.

  2. Compute the gap: D = b - a.

  3. Pick a nearby easy number M close to S (like 100, 200, 500, 1000, 2000…), since M² is easy.

  4. Work out the offset: E = M - S.

  5. Use the formula:

    a × b = (M² - 2ME + E² - D²) / 4
    • M² is easy to square.
    • 2M is easy to multiply by.
    • E² and D² are small.

  6. Finally, just divide by 4 (halve twice).

Example 1: 996 × 1007

  • S = 996 + 1007 = 2003
  • D = 1007 - 996 = 11
  • Pick M = 2000, so E = -3
  • M² = 4,000,000
  • Subtract 2ME = 4000 × -3 = -12,000 → add 12,000
  • Add E² = 9
  • Subtract D² = 121
  • Total = 4,011,888
  • Divide by 4 → 1,002,972

Example 2: 876 × 943

  • S = 876 + 943 = 1819
  • D = 943 - 876 = 67
  • Pick M = 2000, so E = 181
  • M² = 4,000,000
  • Subtract 2ME = 4000 × 181 = 724,000
  • Add E² = 32,761
  • Subtract D² = 4,489
  • Total = 3,304,272
  • Divide by 4 → 826,068

How I found this and some thoughts

I actually stumbled on this while I was messing around with a calculator and trying to find “shortcuts” for doing things in my head. I noticed that whenever two numbers were close together, the product seemed to connect really nicely to their average squared, with just a tiny adjustment. At first I was just doing it with easy pairs like 98 × 102 (which is basically 100² - 2²), but then I wondered if I could generalize it to any pair of numbers. After scribbling on paper for a while (and making a few mistakes 😅), I realized I could always jump to a nearby “easy square” like 1000² or 2000², then just correct it with those little offset and gap terms. What really surprised me is how fast it feels in practice — like you’re just doing one big square you already know, then some much smaller corrections, and finally halving twice. It almost feels like cheating when the numbers line up nicely. I gave it to my dad (who has a degree in maths at cambridge university, the uk one) and he found that when he expanded it out, and then simplified it, it came out as a x b. I hope this formula can help you, I was thinking about keeping it to myself as a secret sauce for making tests with no calculators easier, but I believe in open science, so I’m sharing it with you. Maybe I should write a basic paper on this? Idk, comment below if you think I should.

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