Easily Memorized Multiplication Products

Easily Memorized Multiplication Products

icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM

Feb 17 2012, 08:10 PM #1

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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM

Feb 17 2012, 08:11 PM #2

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icarus
Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM

Feb 17 2012, 08:14 PM #3

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Double sharp
Dozens Demigod
Double sharp
Dozens Demigod
Joined: Sep 19 2015, 11:02 AM

Nov 3 2015, 07:29 AM #4

I would argue for taking μ = 5: this supports what I got when I tried the subitizing game you linked to, and also the results shown on the page (with a steep drop between 5 and 6 objects).

Here's a table for bases 6 ≤ r ≤ 20, as well as 6:10 and 12:10, presuming μ = 5.

In this case there are so many two-digit multipliers in the table with reasonably long lines (like a full 12-times and an almost-full 15-times table, as well as a half-full 24-times table) that I fear it may not be accurate to quantify their difficulty just by their last digit alone. Assuming both digits must be remembered, then only the lines with length shorter than 6, as well as 10 and 20 for being copies of the 1 and 2 lines, can be considered easy, while the lines of 12 and 15 are not. (But 24 is as its length is only 5.)


Tetradecimal and hexadecimal look really sad now! Now octal beats dozenal slightly, due to size being a more important factor: it is in fractions that octal fails. Similarly, it is in efficiency that senary fails. Neither of these factors are covered in this study. This seems to cut the easy-to-memorize tables to {2, 3, 4, 5, 6, 8, 10, 12, 6:10}, presuming we need Me ≥ 80% for a multiplication table to be a viable option for memorization. (This measure does cut out nonary, which is far closer to 80% than any of the other runner-ups. Nevertheless, I think nonary's out because it's odd: 2 is so important that having a difficult-to-memorise line for 2 really cripples a base.)
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Double sharp
Dozens Demigod
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Dozens Demigod
Joined: Sep 19 2015, 11:02 AM

Dec 1 2017, 01:29 PM #5

Given what we've been talking about recently at various threads, such as this one, we should try μ = 9 or 10. Now everything up to nonary or decimal is trivial, everything up to heptadecimal (and enneadecimal) is the same, but with μ = 9, octodecimal now has {0, 1, 2, 3, 6, 9, c, f, g, h} easy, making a figure of 78.9% (171 - 36 = 135), and with μ = 10, vigesimal has {0, 1, 2, 4, 5, a, f, g, i, j} easy, making a figure of 73.8% (210 - 55 = 155).

Tetravigesimal still has only {0, 1, 3, 4, 6, c, i, k, l, n} being helpful, resulting in a much lower figure of 65.0% (300 - 105 = 195). I am not sure how to measure the length of the line as a factor, though. It seems clear that past 20 this is a serious problem.
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