I’ll confine the discussion here to 2 x 2 multiplications. Are there some general rules that can be applied to the suitability of approaches for particular 2 x 2 calculations? What I mean by this is do certain calculations lend themselves to a particular type of calculation. An example would be numbers that are close together can often be handled with the Anchor method, an example being 22 x 27 = 20 x 29 + 2 x 7 = 594. This method doesn’t work so well for something like 37 x 82 which one might use 37 x (80 +2). I try to use difference of 2 squares where possible but unless the numbers lend themselves eg 24 x 36 = 30^2 - 6^2 = 864 then this system doesn’t work so well. What if the units are high, say 67 x 77, is there a method that is best suited to those types of calculations?
For 67 × 77, you can also use the anchor method:
70 × 74 + (–3)(7) = 5180 – 21 = 5159
If you know the 2-digit squares, you can also do the difference of squares (as you already know):
67 × 77 = 72² – 5² = 5184 – 25 = 5159
Some numbers allow different tricks. For example 67:
67 × 78 = 201 × 26 = 5226
67 × 77 = (201 × 26) – 67 = 5226 – 67 = 5159
Or maybe easier:
67 × 77 = (67 × 75) + (67 × 2) = 201 × 25 + 134 = 5025 + 134 = 5159
Or this one. (By factorisation)
67 × 77 => 67 × 7 (11)
=> 449 (11)
=> 5159
well, there is always cross multiplication in your hand if nothing is working.
If the difference between the numbers is small, <20, then the Anker and Midpoint (Diff of Squares) pay off in that you get to calculate with smaller numbers. If the difference is even, I use the midpoint. If it’s odd, I use the Anker. Otherwise I do criss cross.
Thanks for the replies. Criss cross is still the all purpose algorithm and a good fall back, I do wonder if it would be better just to concentrate one’s time on that method and become expert at that. The benefit is you don’t waste time looking for another method and you just launch straight into it. My goal is to be able to calculate any 2 digit by 2 digit calculation in 2 secs or under, basically as quick or quicker than typing into a calculator.
By the way what 2 x 2 do you find is particularly taxing and more time consuming? Is the calculation with high units like 27 x 79, or are there other types that slow you down?
I suspect that’s true, if your goal is speed. I like to vary the methods to avoid boredom which is a real problem for me. I am not training for lightening speed just fluid, accurate performance.
Carries still trip me up from time to time. Especially when there’s a cascade. Digits above 5 tend to induce carries so for this reason higher values are more of a challenge to me.
Memorizing
Really depends how fast you are at seeing patterns, you say you want…
…of course if you’re just straight up fast at criss-cross, we don’t need to discuss the whole issue in the first place, right. For me personally, I got that half a second to simplify the problem, so in the case of…
27 * 79, why don’t you do 27 * 3 = 81 and then 81 * 79
…that one turns out to be really easy, you have 80^2-1^2=6,399… and 8*8=64 everyone knows. Now you just need to divide by 3 again…
6399/3=2,133 … 6, 3, and 9 are pretty easy multiples of 3… so, I don’t know… which is faster, less error prone, easier to calculate? A little algebra goes a long way with the arithmetic.
Now, of course… if you see yourself at some math competitions where you have to calculate calendar dates in under a second to even be competitive, you of course max out your criss-cross skills… but are you really talking about that kind of level for what you’re trying to achieve?
Did you even see that you’re looking at 80^2 i.e., one of the easiest two digit squares since it ends in “0” if you only multiply by 3 and then reverse that afterwards. There’s a lot you can do with just prime factors, etc. when it comes to these kinds of problems.
By the way, I can do 27 x 79 in under 2 seconds using the method I’ve just described. The question is… are you talking about just one problem… or are we talking about being in an actual competition. In that case, you probably don’t want to figure out an easier approach with each problem and just go with one and the same approach, so criss-cross in this case…
Since 37*2=74, you just got 74 x 82 there. So 78^2-4^2=6,068 and just divide that by 2 again for 3,034 and done. I mean unless doubling and halving if such a big deal, then sure… the method ain’t so great… but if you think that’s and issue, I don’t think you’ll be faster using criss-cross. NB: this kinda assumes you got you 2-digit squares memorized in order to be fast.
Hi Bjoern,
Thanks for that suggestion. The methods you’ve described do indeed simplify the calculations and would be faster than the criss-cross method. I’m interested in the strategy that you fast calculators use in the 2 x 2 calculations. Now it may indeed become intuitive over time but I wonder is there a strategy that is used? For example do you think I’ll look for a difference of 2 squares, if I can’t find them I’ll look for an anchor system, if I can’t find that I’ll try factorisation etc.
Is there a hierarchy that can be quickly assessed and if nothing eventuates then it’s the criss-cross method?
Most certainly not. If you watch the fastest calculators in the world, they are calculating 2x2 multiplications in often less than a second before producing a response. When you are going for speed, you do not have time to think about which tricks to use because in the time you figure out what trick to use you have wasted a large quantity of time and the trick would have then been futile. Tricks are used to improve speed but if you have to search for the trick first it would better to just use a predetermined method. Tricks aren’t very commonly used in mental calculation as far as I am aware. I would recommend just practicing criss-cross and building up your speed in that. This will eventually become much faster than any type of trick. I wouldn’t consider myself particularly fast in 2x2, my average time for such problems is around 1.5-2.5 seconds. But you can take Marc Jornet Sanz for example, he is the 8x8 multiplication world record holder. He has trained single digit criss-cross to the extent that it is pretty much subconscious and automatic. You can read a full interview with him on @Daniel_360’s website Marc Jornet Sanz (Interview) – World Mental Calculation.
Thanks Log2, I’m mightily impressed with anyone that can multiply 2 x 2 in 2 secs or less! With the criss-cross method there are at least 4 multiplications to perform and then addition before computing the final answer, how this can all occur in such a short time is staggering. I’ll keep practising my criss-cross and one can only aspire to reach the benchmarks set.
Further to this Daniel, is there a list of numbers that have the capacity for these special tricks?
Any 2 digit number ×11 is adding the 2 digits putting it in the middle and other 2 digits in outside
So 67×77 = 67×11 ×7 = 737×7= 5159
67×11 = 6(6×7)7; carry the 1 voila 737
3 years happened, kinma even liked my post didn’t notice, I didn’t notice either there was a typo there 67×7= 469
Just to extend the info, if you aren’t aware. Not just 2 digits, By 11 you can do this with any digits, also there are ways with multiply by any 1 series.
67 × 7 = 469 (did this one first since it felt easiest not even calculation)
469 × 11 = the trick here isn’t to calculate from the right but actually from the left. any digits that add up more 10 or more are best friends, it’s just a way of distinguish between them.
5 (since 4+6 = 10)
1 (there is 0 here but next two are best friends)
5 (since next there is no best friends)
9
It’s just explaining it taking this many steps, once you get used to it. It’s instant.