What is the most efficient (fastest) multiplication strategy that can be done mentally or with a pencil/paper? We can include strategies that use interesting tools like Napiers Bones or Soroban math. Strategies must generalize well, but answers might include special cases that allow especially quick calculations.
Below are some examples of strategies that are used for the multiplication of integers. I conclude that Soroban mental multiplication is the fastest method based on my research.
23 x 13 =(20+3)x(10+3) =200 + 60 + 30 + 9 =299
Open Array (a graphical version of partial products)
Standard USA Algorithm with Carrying Place Value
"Russian" Multiplication Method Russian Method via Numberphile:
Image below sourced from Popular Mechanic
Image source and description at Wikipedia
Soroban/Abacus Mental Multiplication
I've concluded this method is the most efficient since this is what kids who win the speed contests tend to use (See "Anzan" contests)
Explained at sorobanexam This is a fun article that suggests that the soroban is actually faster than an electronic calculator 4/5 times.
Research (such as Principals to Actions NCTM 2014) indicates building procedural fluency through conceptual understanding is better teaching practice than teaching "fast" procedures alone. This isn't intended for general teaching practice, but rather, for FUN!