# What is the MOST efficient paper/mental multiplication algorithm for integers?

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.

## Partial Products:

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 MethodRussian Method via Numberphile:

Image below sourced from Popular Mechanic ## 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.

## Disclaimer

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!

• I think it depends on the person. Jun 16, 2022 at 23:46
• Efficiency includes not only speed, but also accuracy rate. Generally speaking the faster an algorithm is to execute, the more likely it is that you will make a mistake. Jun 17, 2022 at 1:52
• This post is related. Trachtenberg had some cool tips if general fluency is the goal. matheducators.stackexchange.com/questions/13233/… Jun 20, 2022 at 13:24
• If you know (have memorized) squares of integers high enough (I haven't), then for this particular problem (or any similar small-digit problem whose two inputs have an even sum) the "difference of squares method" (name I made up just now) is worth including -- $23 \times 13 = (18 + 5) \times (18 - 5) = {18}^2 - 5^2 = 324 - 25 = 300 - 1 = 299.$ Note that you can get $18$ by $(23 + 13)/2,$ a procedure (add the two numbers, divide by $2)$ that you can use in general. Jun 24, 2022 at 15:59
• And closely related is the quarter squares method that was widely used for computations until the last 100-some years. Indeed, you can find published tables used for this method on university library shelves, at least those that still have a lot of older math books on their shelves (i.e. not relocated to off-shelve storage somewhere). Jun 24, 2022 at 16:11

If it comes to computers and runtime, I know the Karatsuba algorithm as an example of a very fast algorithm.

If it comes to mental calculation, indeed I think the fun part is that you first need to look at the numbers and then decide on the way to multiply. That then get's very personal and depends on your knowledge. eg. if your fast and accurate at adding, subtracting and taking complements, you might avoid carries with some strategies. If you're fast and accurate at halving/doubling you might use some factoring strategies. For example: in order to multiply by 5 you might find it easier to multiply by ten and then take half. This way, there are neat little tricks and strategies that are powerful, but not in all cases. so again: you learn to look at the numbers and then decide on the strategy. Different strategies are discussed in more detail elsewhere in this forum, e.g. here: Mental Calculation strategies.

• While teaching a PD course that emphasizes fluency (efficient, accurate, flexible) and early numeracy, we were discussing different ways kids develop strategies from inefficient to the most efficient. I had the thought "what is the most efficient strategy?" which naturally led to, "what is the fastest?" Jun 20, 2022 at 13:21

I would say that you need to understand that you are dealing with primary education. This means that, after that, you have secondary education and maybe after that you might have higher education.

The higher the education gets, the higher the number you'll need to multiply.

So, let's try to multiply 12345 times 54321.

Partial products and open array: that makes a total of 25 numbers you need to add.
The Russian method implies that you need to turn one of the numbers into binary format (for 12345, that gives 11000000111001, which implies that, every of the 13 times you're dividing by 2, you risk making a mistake).
Although the Korean style is quite handy to show numbers, I don't understand how you can use that method do to calculations and even if I would understand, I believe that people don't have enough fingers to show the numbers 12345 or 54321 :-)
The soroban technique basically does the same thing as the open array.

So I'd opt for the standard way, as it's used in Europe and apparently also in USA. The others might be fun for showing the children how things were done before, but they are not useful.

• I don’t think “the higher the education, the larger the numbers” is a good assumption. Or if it is, I don’t think there is a real need to calculate exact results mentally… with 2-digit multiplication, you can get to within 2% accuracy on an estimate of 54321*12345 (12K * 55K = 660M ~ 670.6M). Jun 24, 2022 at 14:03
• Just to be clear, this question deliberately is NOT about teaching methods to primary children for the sake of general education. Research summarized in Principals to Actions 2014 strongly suggests that helping students build fluency on a foundation of conceptual understanding (MTP6) is far more effective in multiple ways than teaching "fast methods" or "teaching methods fast." This is a "for fun" kind of question that probably doesn't have a single definitive answer. Jun 25, 2022 at 11:55