This one’s going to be a bit off topic. I have a long piece on the history of pattern matching I want to do but this week is really tight work-wise and this one’s less time-intensive to write.
I’m very interested in “knacks”, or mental tricks that help out in day-to-day life. For example, being able to count quickly and having good memory palaces. Last year I thought I’d memorize a larger multiplication table. Sounds a bit silly, but I come across multiplications a lot, and knowing a large table makes it faster. Like what is 8 feet in inches? Most Americans are only taught the 9x9 table, so they’d have to do 8×10 + 8×2. But some kids are taught the 12x12 table, so they just know that 8×12 = 96.
So there’s clearly some value to knowing a larger table, as it means you can do more stuff in your head, instead of needing a calculator. The only reason we stop at 9x9 (or 12x12) is because you can only do so much with kids before they lose interest. But as an adult I’m willing to put the time in to multiply better. Why not memorize the 99x99 table? Then I can see, say, 14×87 and instantly know it’s 1218.1
One teensy tiny problem: multiplication tables scale quadratically. The 9x9 multiplication table has 45 terms.2 The 99x99 table has 4950. I’m willing to put time into this project, but not that much time.
But knowing something beyond 12x12 would be helpful, the question is just where the cutoff is.
Let’s consider why knowing a larger table is useful in the first place. If you don’t memorize 14×87, you can quickly compute it by decomposing the equation into four unit multiplications. That’d give you (4×7) + (4×80) + (10×7) + (10×80). Individual multiplications are pretty easy, as is addition, so memorization doesn’t save you much calculation time.
My problem is instead working memory. I can’t keep all of the intermediate results from four multiplications and three additions in my head. Once I get confused I have to throw away all my work and start over. I have to repeat
28 + 320 three or four times, and that’s what makes mental multiplication so slow and error-prone.
We can also decompose 14×87 into (14×7) + (14×80). That’s two multiplications and one addition, which is easier to keep in my head. Even though multiplying with the 9x99 table takes more working memory than the 100x100 table, it’s significantly less than using the 9x9 table. And unlike square tables, 9x99 scales linearly: I’d “only” need to learn about 750 new terms.
That’s doable in a few months of light work. I figured I’d start by learning 13x9, then 14x9, and gradually work my way up.
It’s been 20+ years since I last studied multiplication tables, but I vaguely remembered doing a lot of drills, like 9×3, 2×7, and so forth. Since vague memories of 20-year old techniques are obviously the best way to do anything, that’s what I went with. I grabbed a math worksheet generator and did about 100 problems a day. Odd days I’d drill a new number, even days I’d do review. Over time I wouldn’t need to solve the problems, I’d just know the answers, and that would expand my table.
This sucked! It was slow and dull and I’d constantly forget old entries. I got as far as 9x18 before I gave up, and within two weeks I’d forgotten everything. Complete waste of time.
I recently got the bug again but wanted it to actually stick, so I swallowed my pride and looked into how people actually taught times tables.
There’s two groups of people who care about teaching multiplication. The first is elementary school teachers, who only care about squishy kid brains. Their audience is undisciplined and easily bored, and their only goal is to get kids to memorize a 9x9 table. The other group is the mind sport enthusiasts. They want to be able to multiply 5+ digit numbers as fast as possible. They do this by mastering a wide set of tricks, some of which can take years to master.
And they get to write down each digit as they calculate it, significantly reducing the working memory load and making the techniques completely useless to me.3
That’s the annoying thing about learning knacks as an adult. There’s a lot of material out there for children and sometimes stuff for hardcore competitive types, but never anything for adults who just want to make their everyday lives slightly better.
Still, I researched both groups, and found a potentially useful technique in the kid camp: skip counting.
Skip counting means counting along a number line but only saying the numbers that fit in whichever sequence you’re working on. To skip count in fives, you say 5, 10, 15, 20, 25, 30…. […] Before we can recall individual times tables elements and answer random times tables questions in our head, we need to learn the skip counting pattern of each times table.
While the goal is a random-access table, you first teach the student a linked-list: 3 → 6 → 9, etc. It’s easier to remember the numbers as part of a sequence. In fact you can break the list into two four-element subsequences, digits 2-5 and 6-9, and memorize those separately. That makes adding a row to the table more manageable.
I wrote out the two skip lists for 13, like this:
26 39 52 65 78 91 104 117
Seeing it that way, I realized that I could improve on the skip list, with the power of spatial memory.
There is one knack that’s widely available to everyday adults: memory palaces. Humans have incredible spatial memory, and it’s much easier to remember things oriented in a space. I wondered if I could use this to memorize multiplication tables. Here’s all the digit multiplications for 13, laid out in a 3x3 square:
13 26 39 52 65 78 91 104 117
Now we’re not just memorizing numbers, we’re memorizing a shape. Instead of remembering that 13×7=91, I remember that 91 is in the lower-left cell of the grid. The grid layout also opens new memory heuristics. Here’s just the one’s digits of the 13 grid:
3 6 9 2 5 8 1 4 7
And here’s the one’s digits of the 91433 grid:
3 6 9 2 5 8 1 4 7
Each digit has a “fingerprint” you can use for error-checking. Is 13×8 104 or 102? 8 is in the bottom center cell, so the last digit must be a 4, so it can’t be 102. You get the same digits with regular skip lists, but the column patterns are a lot more obvious this way.
So far this has been working a lot better! Some misc thoughts:
Originally I didn’t think of multiplying by 1 as being relevant, since it’s obvious that 1×n = n. But if you consider it a valid element, you can lay out each row as a 3x3 square. Otherwise you have to use a 2x4 rectangle, which is less spatially memorable.
The grid method opened up fingerprinting as a technique. But there were also some neat heuristics I noticed while brute forcing, like for p > q, np×q = nq×p - 10n×(p-q). ie if 13×7=91, then 17×3=41. If 23×7=161 then 27×3=81.
Okay this already took twice as long as I expected to write, so whoops project “don’t spend too much time on the newsletter this week” was a miserable failure ¯\_(ツ)_/¯. Have a great week everyone!
This is especially useful because it also opens up decimal multiplication. If you know that 14×87 is 1218, then you also automatically know that 1.4×8.7 = 12.18. This comes up a lot in metric conversions: miles to kilos is 1.6, kilos to pounds is ~2.2, etc. ↩
Multiplication is commutative, so if you know the upper half + squares, you know the whole table. The number of distinct figures is (n²+n)/2. In practice it’s a little less than that, because you don’t have to worry about the 1-row. ↩
Okay I’m exaggerating a little, cross multiplication seems pretty nice if you can write stuff down, but if I have pencil and paper I’ll also have a calculator, so… ↩