General Squaring (Duplex Method)

The Duplex Method, an application of the "Vertically and Crosswise" sutra, is a universal tool to find the square of ANY number, big or small!

What is a "Duplex"?

The "Duplex" (D) is a simple operation depending on the number of digits:

• For 1 digit (a): D(a) = a²
• For 2 digits (ab): D(ab) = 2 × a × b
• For 3 digits (abc): D(abc) = (2 × a × c) + b²

Example: Square 123

1. Step 1: Calculate the Duplexes.
   • D(1) = 1² = 1
   • D(12) = 2 × 1 × 2 = 4
   • D(123) = (2 × 1 × 3) + 2² = 6 + 4 = 10
   • D(23) = 2 × 2 × 3 = 12
   • D(3) = 3² = 9
2. Step 2: Combine the Results.

   Our results are: 1 | 4 | 10 | 12 | 9

   • Last digit is 9.
   • Next is 12. Write 2, carry 1.
   • Next is 10. Add carry: 10 + 1 = 11. Write 1, carry 1.
   • Next is 4. Add carry: 4 + 1 = 5.
   • First digit is 1.
3. The Answer: Reading the bold digits, we get 15129.