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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.
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.
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 “antireason”, 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 eightynine”; and then that “eightynine” 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!
Later, when I was learning the products up to 1500, I discovered a second antireason, also caused by my tendency to group digits in pairs. Since I read 1343 as “thirteen fortythree”, 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 antireason, 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 bruteforce 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

See Also@ November (2004) o April (2009) 