What Is Memorable?

Now that I've told you more than you wanted to know about associative hooks, let's get back to the main point, how I remember products.

Actually, first, let me turn the question around. I remember any particular product because it's memorable for one or more reasons; the right question isn't how I remember it, but rather what makes it memorable (to me) … what's the reason that it's memorable?

Originally, I thought I'd just point out a few interesting reasons I'd noticed, but then, when I sat down to write the essay, on a whim I decided to do a complete survey of all the products, to see if there were any reasons I'd missed. There were, of course … no surprise there. But, there was a surprise: I'd expected to find a cloud of vague, overlapping reasons, and to throw most of them away as useless; instead I found just a few snowflakes, crisp and distinct. So, now I can tell you all about every one of them.

First of all, a product might be memorable because one of the numbers involved is memorable. The number might be a factor, in which case we're back to talking about strong associative hooks for primes; or the number might be the product itself. And, in the latter case, just as in the former, it doesn't matter if the number is memorable for a good reason or a stupid one … 511 = 7×73 is memorable because it's a Mersenne number; 247 = 13×19 is memorable because it reminds me of the jargon “24/7”, meaning 24 hours a day, 7 days a week.

Or, a product might be memorable because of some relationship between the numbers involved. I'll divide the relationships into categories; they're all pretty stupid, but I can't help that.

• Sometimes the factors have digits that match. Usually it's the last digits, as in 559 = 13×43, but it can also be the inner digits, as in 713 = 23×31, or even the inner and outer digits, as in 403 = 13×31.
• Sometimes the last two digits of the product can be transformed to make the second factor. In 581 = 7×83, for example, the “81” becomes “83”. The leading digit often stays fixed, but not always … there's 679 = 7×97, where the digits swap, and also the most complicated one, 871 = 13×67, where the “71” decreases to “67”.
• Occasionally, the factors concatenate nicely. I only have two examples of that, 623 = 7×89 and 481 = 13×37. The latter may only be clear to the elite, but to me it was so memorable that it carried over to 851 = 23×37.

I misspoke, above, when I said that any product is memorable for a reason … there are a few products I remember for no reason at all, as far as I can tell.

• Some I've known too well, for too long, to be able to analyze. Within that category are the eight squares, the one cube, and the products 77 and 91.
• Some I just absorbed instantly and easily, for no apparent reason. That mostly happened early in the process, when the numbers were small and my mind was fresh, but it did happen occasionally all the way to the end, for example, with 901 = 17×53.
• And, conversely, some I just couldn't get a handle on, and was only able to learn after much repetition. The very worst one, I think, was 731 = 17×43.

Now let me go back for a second and repeat something I hinted at before, which is that a product can be memorable for more than one reason. The product 511 = 7×73, for example, is memorable not only because it's a Mersenne number, but also because the inner digits of the factors are the same. Still, in practice, there's usually one reason that's much stronger than the rest.

Finally, when I was learning the products, I noticed a single “anti-reason”, a thing that makes products less memorable. As I said earlier, in Four Digits, I tend to group digits into pairs. So, for example, when I was trying to remember that 689 = 13×53, I'd always think of the left side as “six eighty-nine”; and then that “eighty-nine” would associate to other products that ended in the same two digits, notably, 589 = 19×31. Then I'd get the factors all mixed up!

• I like the computer analogy here (see Operations). An association is like an entry in a hash table; the root of the problem is that my mind wants to store the products using only the last two digits as the hash key.
• I had particular difficulty whenever a product in the 800s collided with one in the 500s. I think it was the visual similarity that did it; that goes against what I said about my memory being verbal, but then, I was reading the number off a flash card; maybe somehow the mistake happened in reading, not in memory.
• Fun trivia facts: there are no three-digit products that end in “57”, and five that end in “89”. All other suffixes are intermediate between those two.

Later, when I was learning the products up to 1500, I discovered a second anti-reason, also caused by my tendency to group digits in pairs. Since I read 1343 as “thirteen forty-three”, it's easy to confuse the product 1343 = 17×79 with the product 13×43 = 559. The confusion isn't as bad as with the first anti-reason, since the association leads to a product, not to factors, but it still could be a problem in the future.

How many confusing products are there? Well, there are 22 primes between 7 and 100, and for each ordered pair (p,q) we can construct a domino with left side p &₂ q and right side p×q. For example, (13,43) yields 1343 : 559 while (43,13) yields the flipped domino 4313 : 559. When two dominoes fit together, the middle number is a confusing product. A quick brute-force calculation shows that there are 23 ways to fit two dominoes together (plus 22 with the left domino flipped—not 23 because one way has left domino 7373 : 5329), so there are 23 confusing products. There are also two (plus two) ways to fit three dominoes together, but that's as far as it goes.

3753 : 1961 : 1159 : 649
4397 : 4171 : 2911 : 319