· Portal  Help Search Members Calendar 
Welcome Guest ( Log In  Register )  Resend Validation Email 
Welcome to Dozensonline. We hope you enjoy your visit. You're currently viewing our forum as a guest. This means you are limited to certain areas of the board and there are some features you can't use. If you join our community, you'll be able to access memberonly sections, and use many memberonly features such as customizing your profile, and sending personal messages. Registration is simple, fast, and completely free. (You will be asked to confirm your email address before we sign you on.) Join our community! If you're already a member please log in to your account to access all of our features: 
icarus 
Posted: Feb 17 2012, 08:10 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Easy Multiplication ProductsThis related thread quantifies the numbertheoretical relationships of products in a multiplication table to the number base r. The previous, geeky thread laid out the ultimate objective for such a study: quantify the ease of memorization of a given table. I recognized in that thread that there are two main components in the problem; human cognitive ability and elementary number theory, that the latter seems “less difficult” to quantify. The multiplicationorder thread addresses number theory, but it is clear that it doesn’t say everything about how easy it might be to memorize a multiplication table. I am an American who learned multiplication in 2nd and 3rd grade (ages 78, between 1977 and 1978). In that era, rote memorization of product lines was the rule. We would memorize “the fives”, i.e., the line of products with five as the multiplier: {5, 10, 15, …, 50}, at first through 10 as the multiplicand. We then learned the elevens and twelves, so that ultimately we memorized a table with 78 unique products, rather than the strictlydecimal table of 55 unique products. This is because the eleven product line is nearly trivial {11, 22, 33, …, 99, 110, 121, 132}, and the dozens are, as my aunt put it “really very important, twelve is a very useful number”. This method of quantifying “easilymemorized” facts will use the following definitions. We will not stick to the numbertheoretical classifications of digits and relationships with base r, though these are important because we can calculate the “ease” using these. We will need to presume some things about human cognition, because I am not a researcher / cognitive scientist, and don’t have the benefit of a university with grants, etc. to build a proper study. (But this would be a cool thing to study, if it might be studied). We need some basic definitions so we can talk about the subject. Let the integer r ≥ 2 be a number base. Let the nonnegative integer n < r be a digit of base r, with the digit “0” representing congruence with r. (We will ignore the special case where “0” signifies actual zero, i.e., when the placevalue “0” stands alone.) Here are the “rules” of the quantification of “ease” of memorization of a multiplication table in base r: We need two metrics. The first is numbertheoretical, but can be observed without resorting to theory (however I have a way to calculate these and may post it later). The second relates to human cognitive ability and is based on a notion I think would govern how much a person can keep in mind while calculating. Yes, it’s a guess, and the value of this number would alter the calculations. Let the positive rational number λ be the “period length“ of the cycle of unitplacevalues in a product line for digit n in base r. I will assume that product lines of any digit n of base r with an integer λ, whether the unitplacevalues increase or decrease with an increasing multiplicand, will be “easy” to memorize. Examples include digit 2 base 10 {0, 2, 4, 6, 8, 10, 12, …} and digit 8 base 10 {8, 16, 24, 32, 40, 48, …}. I will assume that any product line of any digit n of base r with a noninteger λ will be opaque to pupils, and not be “easy” to memorize. Example of this case is digit 4 base 10: {4, 8, 12, 16, 20, …}. For digit n = 2 of base r = 10, λ = 5. For digit n = 4 of base r = 12, λ = 3. (For any n such that n  r, λ = r/n.) Let the positive rational number μ be the “mnemonic capacity“ of an average human being. We will suppose this is equivalent to the maximum subitizing range of 7. (For more about subitizing, search it, or take a look at this neat test at Wolfram). The human ability to subitize, that is, reliably quantify without counting, falls between 4 and 7 and a cogent study would be in order to really define this variable. I use 7 because this length had served the longest local telephone number in the United States, studied by the American phone monopoly a while back. A more solid value could be 6; for some it might be 4. Let’s presume μ = 7. If any product line has λ > μ, we will deem it “difficult” to memorize. Conversely, any product line with λ ≤ μ will be deemed “easy” to memorize. We will universally declare any product involving the trivial divisors {1, r}, common to all bases r, trivial to memorize. The problems “1 × x” and “10 × x” would seem to present little challenge to compute, no matter the size of the number base r. We will universally presume any product involving the largest digit, ω = (r − 1), “easy” to memorize. The effect of presuming μ = 7 implies that the entire multiplication tables for bases r ≤ 7 are “easily memorized”. Generally, the entire multiplication tables for bases r ≤ μ are “easily memorized”. 
icarus 
Posted: Feb 17 2012, 08:11 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Easily Memorized Product MapsHere are a few “easily memorized product maps” in several small bases r that we always find ourselves talking about. Since we presume μ = 7, we won’t analyze bases less than 8. Let’s look at maps for bases {8, 10, 12, 14, 15, 16}. In the maps, purple signifies “trivial products”, red “easy products”, and muted colors “difficult products”. The “difficult” beige and the gray products represent those that are regular or semicoprime, and those that are coprime, respectively.


icarus 
Posted: Feb 17 2012, 08:14 PM


Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Quantifying the Ease of Memorization of Multiplication TablesLet’s quickly quantify the population of “easilymemorized products” for bases 8 ≤ r ≤ 20. Suppose x is an arbitrary positive integer. Let the triangular number function Tri(x) = x(x + 1)/2. Let the integer e be the number of qualifying digits in base r. Let M_{e} be the quantity of “easilymemorized products” in base r: M_{e} = er − Tri(e − 1) Let the positive integer M be the total population of unique products in the multiplication table of base r: M = Tri(r) Now we can produce the following table for the bases considered:
Thus, it is clear that the decimal multiplication table is rather “easy to memorize”. Only dozenal and octal exceed the “ease of memorization” of the decimal multiplication table for r > 7. (All r ≤ 7 are trivial to memorize.) The hexadecimal table is significantly more difficult to memorize, on par with the tables for bases {14, 15, 18}. Further, the computations suggest that the duodecimal multiplication table is the easiest table to memorize outside of bases r < 7. It is also the largest of {2, 3, 4, 5, 6, 7, 8, 10, 12}, meaning that we maximize our human computational capability when we use base twelve. With only three “difficult” products to memorize, a case could be made that the three holdouts could be tackled in a lesson, overcoming the “difficulty”. All this is written based on an assumption; thus if the actual value of μ is different, or if we deem sequences like that of digit 4 base 10 {0, 4, 8, 2, 6} “easily memorized”, or the multiplication facts associated with the multiplier 2 in any base as “easily memorized”, our results will be different. (Note that in the table above, I gave digits 2 and e base 16 the “benefit of the doubt”, and included them in the total, though their values of μ disqualify them.) 

Double sharp 
Posted: Nov 3 2015, 07:29 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
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 twodigit multipliers in the table with reasonably long lines (like a full 12times and an almostfull 15times table, as well as a halffull 24times 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 easytomemorize tables to {2, 3, 4, 5, 6, 8, 10, 12, 6:10}, presuming we need M_{e} ≥ 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 runnerups. Nevertheless, I think nonary's out because it's odd: 2 is so important that having a difficulttomemorise line for 2 really cripples a base.) 

Double sharp 
Posted: Dec 1 2017, 01:29 PM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
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. 