Embarrassingly for someone who works in primary schools, I’ve never really nailed these - even up to 12x12 there are some that I need to think about.
I’m keen to tackle this at some point and go beyond it - perhaps up to 20x20. I’d be interested to know if anyone’s had success with mnemonic techniques in learning their times tables?
I learned them by rote when younger, but they’ve faded now, so most of them aren’t instant, and I’m not always confident it’s right until I check it. I toyed with what it would take to learn the 100x100 tables, but my estimate was that I’d need to learn 4,851 products ((100)(100 + 1)(1 / 2) - 100 - 99) and with both 3-digit and 2-digit systems would need 9,902 images ((4,851 * 2) + 200) spread across 5051 loci (4,851 + (100 * 2)) placed on at least 2 sq. ft. of surface area of a lukasa, which is not undoable, but doesn’t seem worth it to me
Since tracing the multiplier/multiplicand column and row to the product could get unwieldy for numbers further from the edge of the table, I thought that visually differentiating the loci could provide a good source of orientation. I thought that using black objects by default for product loci, white objects for a product locus if both the multiplier and multiplicand are prime numbers, blue objects for a product locus if the product itself is a highly composite number, and light blue objects for a product locus if both the multiplier and multiplicand are prime numbers and the product itself is a highly composite number. I estimate that this would produce a good distribution of non-default-color loci to make different areas of the lukasa feel different even when looking at only a small part of the board
I suspect that that sort of approach would not fit the usual usage case as well, though, where recall speed is more favored. Perhaps combining the Japanese kuku with mnemonic images would work? (12 ^ 2) * 3 == 432 images with a 3-digit system (less if you skip the ones and tens; only (12 ^ 2) == 144 images if you only encode the product as an image but associate a kuku-like phrase with that locus), with the idea that by repeating the multiplier and multiplicand at the same locus as the resultant product, it would be easier to access quickly?
I used to use spaced repetition (e.g. Anki) with my coaching clients, but now I have some more targeted exercises. Using spaced repetition, consider each fact “known” if you answer it immediately. So if Anki asks you “6×8” and you’re like “oh yeah after 40 it’s 48” rather than immediately knowing it’s 48, then I’d consider that as not yet “known” for the purposes of Anki.
There are only 55 facts to learn (ignoring the 1× and 10× tables and duplicates e.g. 6×8 with 8×6) and you’ll know most of them already, so it won’t take long to drill them.
Regarding tables up to 20, I generally recommend to learn up to 19×9 but not all the way to 19×19. The former is much more useful, because you can use them for splitting up larger multiplications. For example, 57 × 17 = 850 + 119 = 969 if you know your 17-times table. You don’t have to keep the area that you learn a square 
Regarding tables up to 99×99, I agree that this takes a lot of work. I count 4005 multiplication facts (ignoring 1×, 10×, 20×, …, 90× tables and duplicates) with an average length of almost 4 digits, so about 15000 digits. Unless you want this as a specific memory challenge, I consider it very high-hanging fruit in the context of mental math.
That’s very helpful, thanks. I like the idea of going up to 9x19.
I think you’re right that it’s gotta be Anki, maybe using some word/image associations too to make it more fun and hopefully nail them quicker.
I’ll try up to 99x99 as soon as I learn that I have that many years to live! 
There are also some discussions here: 1x99 Multiplication Table - #2 by Josh
If your table is that large, be sure that the kitchen in your memory palace can accommodate it.
I really don’t want to deny anyone their methods, but I think that learning multiplication tables by heart is a complete waste of time. Once you have practiced multiplication two-digit by two-digit enough, you can recall it just as quickly or faster than memory contents of this size. In addition, you have laid the foundations for a deeper understanding of calculations. Even after a long break, you can get back into it much faster than refreshing the multiplication tables again. Please don’t get me wrong, I also have a very large number of numbers in my memory. But at some point I specialized only in values that cannot be calculated particularly quickly and can therefore be used as a tool for other calculations.
Speed Math for Kids by Bill Handley has my favorite method!
No memorization needed, either! Take that memorizers! 
It’s all calculation done on the fly. Very cool.
I showed it to my 12 year old son, we were mentally calculating 2 digit by 2 digit numbers (98 x 96, etc) within 30 minutes.
What I really dig is that it got me much more interested in ways you can play with the basic properties of addition and multiplication.
Out of interest, which type(s) of method was he recommending?
While I prefer the “creative” methods for 2×2 multiplications, each method has its pros/cons. I’m just curious which approach you/the book recommends as being most “cool”/“interest[ing]” and “for Kids”. As in some cases, that’s the most important feature!
I thought this might interest you, Daniel 
Handley doesn’t start out saying this (obviously, how boring that would be for kids) but I got curious and worked out that his method is a creative use of the property of multiplication over addition.
It works for multiplying any size digits.
Lets start with one-digit numbers:
7 x 8
Pick a reference number, both numbers are pretty close to 10, so 10 is the ref number.
Subtract one factor from your reference number.
10 - 8 = 2
Do the other:
10 - 7 = 3
Write (eventually visualize) 2 below the 8, and circle it 
for funsies, same for the 3 below the 7.
7 x 8
(3) (2)
Pick one of the circled numbers (doesn’t matter which, it ends up being the same result) and subtract from its caddy cornered factor.
We’ll do 2 from 7 in this case:
7-2=5
And multiply that result by your convenient reference number.
Then say that out-loud so you can beat those punks using calculators and make it look like you’ve already got the calculation just about finished before they are finished number punching.
7 - 2 = 5 (x RefNum: 10 = 50)
“Fiftyyyyyy…”
Now just multiply your two circled numbers and add that result to the number you’re almost finished speaking
(2) x (3) = .....
“…yyy…six!”
Quick Note:
This example, though short, has been, admittedly, quite a pain to write (if this sorta thing is easy for you, congratulations, ask Josh to kindly direct you to the bragging section of the forum to share your ease).
So I’ll wrap up even though undoubtedly questions are likely to abound like “Well how would you…”
lets do 11x 13
Same thing. Reference number is 10 again because its easy.
How do you get 11 from 10? add (1)
How to get 13 from 10? add (3)
(1) (3)
11 x 13
^the numbers go on top this time, indicating positivity
13 PLUS the caddy cornered (1) is 14
times your reference number: 10
"One hundred fortyyyy......."
While saying that, quickly multiply 1 times 3 and add to the result
"yyy..... Three!"
7 times 12:
RefNum = 10
+ (2)
7 * 12
- (3)
---
Result = (7 + 2 = 9 OR 12 - 3 = 9) * RefNum # 10
print("Ninetyyyyyy........")
AddToResult(2 * -3) = -6
print("yyyy....EightyFour") # they didn't even notice the correction! Noiceeeeee
Works for larger digits by changing the reference number.
I’ve mentioned before in the forum that I prefer in person conversations over writing forum posts.
Mainly it’s because I personally find writing these explanatory posts both challenging and time-consuming. That said, I’ve been writing carefully crafted explanations (to varying degrees of success) for literally decades.
Not because I enjoy it, but because I appreciate reading well crafted explanations.
However, and I’ve never mentioned this before on this forum (or online at all), both my son and I have strong cases of ADHD.
I didn’t realize for so very long the cognitive challenges this presents that 96-ish% of the general population does not have to deal with. I just went along my merry little life for over 36 years doing what I felt I had to do to come to adequate conclusions, very much avoiding comparing myself to anyone on the cognitive front. And frankly not even “believing” in ADHD.
Until, of course, the seemingly endless amount of decades of research on the subject was presented clearly to me (not that long ago).
If I seem cranky, I am.
To discover not only that something you scoffed at the possibility of existing for years actually does in fact exist—and, at the very same moment of realization, to suddenly face the reality that you have the previously considered ridiculous condition… while the grand majority of the population does not…
Well. It’s recent enough that the realization’s bad “aftertaste” might not even actually be in the “after” stage yet.
On this forum, I avoid sharing anything that I haven’t tried and used. I break the rule occasionally, but i typically find that sort of hypothetical mental masturbation generally unhelpful and thus avoid doing it myself.
This technique worked for me and my son. And rather quickly, I’ll add. And is extensible, as I was multiplying 3 and 4 digit numbers together in my head.
If anyone wants more details on it, pick up Bill Handley’s book. I check mine out from the library (digitally, at that).
If anyone has doubts or questions, take it up with Bill. I’ve got so many other things on my plate. I won’t be arguing for or against any methods here.
I’m sharing what worked for a father and son with (now very apparent) cognitive impairments.
I already, much to my own chagrin, upset someone I care about because I took so much time writing this little post.
Good luck on the math stuff, y’all.
Enjoy and have fun! For me the enjoyment and fun is the best part!
Many kind regards,
Beau
Thanks for the explanation—that is clear!
I’d count this as one of the “creative” methods, which—while working well only for certain multiplications—is much nicer in these cases than trying to brute-force it using a general-purpose method. And therefore perhaps more satisfying. Good to know what is interesting from a general perspective.
What is also nice about this method is that (apart from the mental calculation side) it also introduces some concepts from algebra without overwhelming some people with variables and fairly formal-looking notation.
Taking your 11 × 13 example, it generalizes (as you know) to:
(10 + something) × (10 + something else) = 100 + (add both somethings) × 10 + (multiply both somethings).
And the 96 × 98 example as:
(100 – something) × (100 – something else) = 10000 – (add both somethings) × 100 + (multiply both somethings)
In formal algebra, this might be written as (100 – a)(100 – b) ≡ 100² – 100(a + b) + ab, but for many students that might feel like a robotic manipulation of symbols and rules. But the mental calculation shortcut is a more accessible way to see the same. And for us at least, “fun”.
Oh I like your breakdown! Thanks!
The other aspect I like about it (along with the points you mentioned!)…
lord almighty how adhd is that?
Rereading that I wrote this after you just wrote:
– Time for a break from the keyboard.
…is that it also gets my son and myself mentally practicing basic arithmetic operations in a non-overwhelming and purposeful manner.
Merely doing a little addition and subtraction with an extra helping of multiplication—all mentally in pursuit of the greater result.
Hats off to Bill for a clever approach, methinks!
Check this book ‘trachtenberg system’