2313: Wrong Times Table

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Wrong Times Table
Deep in some corner of my heart, I suspect that real times tables are wrong about 6x7=42 and 8x7=56.
Title text: Deep in some corner of my heart, I suspect that real times tables are wrong about 6x7=42 and 8x7=56.


A "times table" (or multiplication table) is a table used to show the products of numbers. Typically, elementary school children are taught to memorize the table of whole numbers up to 10 as part of learning arithmetic.

In this comic Randall supplies his own alternative version of the multiplication table, with entirely incorrect values that nonetheless "feel" reasonably correct to him. It is unclear how his values are derived, as they don't follow a consistent pattern, but it could be that when calculating products, he sometimes has to correct his mental arithmetic, perhaps thinking along such lines as "8*4 is 36... Or, wait, is it 32?". Most of the values are transposed from their correct position (e.g., adding or subtracting one -- or two, or three -- from one or both multiplicands), some are "off by one" (or two, or by a factor of two), and some (mostly in the 1 row and column) could be created by adding, subtracting, or dividing the two factors instead of multiplying them. It is notable that some properties of mathematics are not followed, as sometimes smaller multiplicands multiply to a larger product than larger multiplicands, and sometimes two even multiplicands produce an odd product.

The table is symmetric, indicating that Randall's form of multiplication is commutative.

The title text (referencing Randall's suspicion that 6x7=42 may be wrong) is an allusion to The Hitchhiker's Guide to the Galaxy, in which the ultimate answer to life, the universe, and everything is said to be forty-two. However, in the book this answer is meaningless without knowing the ultimate question, and so to calculate the ultimate question, a planet-sized computer is constructed. This later becomes Earth, but Earth is destroyed shortly before its calculation is complete. Arthur Dent, one of the last surviving humans, has some white mice (pan-dimensional beings looking like white mice to us) try to get him to give them his brain, so they could attempt to recreate the ultimate question, hoping it may be stored within his brain since he was part of the computer matrix up to just before Earth was destroyed a few days before completing a 10 million year calculation. Arthur refuses, and the mice try to think of some question that makes the answer 42 make sense, like "how many roads must a man walk down". They also suggest 6x7. Arthur later tries to recreate the question himself by picking letter tiles from a bag, and produces the sentence "What do you get if you multiply six by nine". This leads him to remark "I always thought something was fundamentally wrong with the universe." Note, however, that operation of said planet-sized computer was disrupted, both by its near-total destruction and by the much earlier crash-landing onto it of the 'B' Ark and its somewhat useless passengers, so it's also possible the universe is okay and only the question was computed incorrectly. As it happens, 6x9 = 42 in base 13, but Douglas Adams has disclaimed this as being a mere coincidence. In Randall's table, neither 6x7 nor 6x9 are said to result in 42, but 7x7 is.

If we consider the smaller multiplicand to be a and the larger to be b, then (one of infinitely many possibilities of) the formulas used by Randall are as follows:

1 2 3 4 5 6 7 8 9 10
1 a-b a/b a+b a+b a+b a+b a+b a+b a+b b-a
2 a/b 2ab a+b a(b-1) a(b+1) a(b+1) a(b-1) b+10 b+10 a(b+1)
3 a+b a+b ab+1 (a+1)b a(b-1)+1 (a-1)b a(b+1) (a+1)b a(b-2) a(b+1)
4 a+b a(b-1) (a+1)b 2ab (a+1)b (a+1)(b-1) ab+1 a(b+1) a(b-2) a(b+2)
5 a+b a(b+1) a(b-1)+1 (a+1)b 2ab (a-1)b a(b+1) a(b+1) a(b-1) (a+1)b
6 a+b a(b+1) (a-1)b (a+1)(b-1) (a-1)b (a-2)(b+2) a(b+1) ab+2 a(b+3) a(b+2)
7 a+b a(b-1) a(b+1) ab+1 a(b+1) a(b+1) a(b-1) (a-1)(b+1) (a-1)(b+1) a(b+2)
8 a+b b+10 (a+1)b a(b+1) a(b+1) ab+2 (a-1)(b+1) a(b-2) ab+2 a(b-3)
9 a+b b+10 a(b-2) a(b-2) a(b-1) a(b+3) (a-1)(b+1) ab+2 a(b-1) a(b-1)
10 b-a a(b+1) a(b+1) a(b+2) (a+1)b a(b+2) a(b+2) a(b-3) a(b-1) a(b+1)

The correct multiplication table for the numbers 1-10 is below:

× 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100


[Caption above table:]
Wrong Times Table
The incorrect answers that feel most right to me
1 2 3 4 5 6 7 8 9 10
1 0 ½ 4 5 6 7 8 9 10 9
2 ½ 8 5 6 12 14 12 18 19 22
3 4 5 10 16 13 12 24 32 21 33
4 5 6 16 32 25 25 29 36 28 48
5 6 12 13 25 50 24 40 45 40 60
6 7 14 12 25 24 32 48 50 72 72
7 8 12 24 29 40 48 42 54 60 84
8 9 18 32 36 45 50 54 48 74 56
9 10 19 21 28 40 72 60 74 72 81
10 9 22 33 48 60 72 84 56 81 110

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I wonder if there is such a mathematical expression which follows the definition of "multiplication" as in advanced calculus which actually provides the results on the table; i.e, some sort of bijective homomorphism that maps v: V*V --> V

Such an illogical table. Smaller numbers multiply to larger answers than larger numbers? Even numbers multiply to odd numbers?! How?!?! What sort of illiterate alien declared this to be the multiplication table?! 20:54, 29 May 2020 (UTC)

Feels like an element of logarithmic thinking, like the idea that the space between 1 and 2 is greater than between 99 and 100, or that 3 is halfway to nine. ancepsinfans 21:24, 2 June 2020 (MSK)

This is easily one of the worst XKCD comics, period. Not funny, nor clever. Just seems like noise. 20:57, 29 May 2020 (UTC)

I suspect Randall may have just been feeling random, perhaps after several months of mostly Coronavirus-related comics. Barmar (talk) 21:13, 29 May 2020 (UTC)
That's fair, I'm being a bit harsh, but this just comes across as exceptionally meaningless and contrived, so much so that I felt the need to come here and comment immediately for the first time ever 21:18, 29 May 2020 (UTC)
I relate to certain mathematical facts not sounding right, like how 54 intuitively feels like it's divisible by 4. Nonsensical, but makes sense anyway. 21:42, 29 May 2020 (UTC)

This seems like the multiplication equivalent of looking at a word and thinking it is spelled incorrectly. Sometimes I look at a simple word like "fish" and think: "That can't be right." Sometimes multiplication can feel that way, particularly 7's because those were tricky for some reason. The alt text confirms fishiness with 7's 21:09, 29 May 2020 (UTC)

Is it weird that I don't get this? I have this sense of "that is obviously wrong" when it comes to multiplication of small numbers like these, even with sevens and eights. If I read that 7 * 8 = 54, my brain screams "NOOOOOOOOO IT IS 56 YOU IDIOT!". 21:14, 29 May 2020 (UTC)
It's lucky for you! Careless errors [of all types] can be annoying, and sometimes difficult to locate... Some of us have ingrained this information better than others. (This comic seems less like a joke and more sharing a hindrance Randall suffers from when doing arithmetic. And speaking personally, I can certainly relate to that.) 18:05, 1 June 2020 (UTC)
I wouldn't say it's weird, but maybe you don't get it because you're better at the memorisation or multiplication that we were taught in school. Try spending a few hours working in base nine or base thirteen, you might see how you commonly make the same errors. I know that I have difficulty with multiplying even small numbers, so I take longer to do it so I infrequently make these common errors. But I do like to work in other bases occasionally so I can vaguely relate to the subject matter of this comic. 18:20, 2 June 2020 (UTC)
I never fully memorized the times tables, to this day (50+ years later) I use tricks... 7*6? Well, I know the 5's, and 7*5=35, so now add another seven.... 42? I've always preferred to understand how to get to an answer than to memorize the answer, for some reason. Stuff I use frequently or that are especially useful (i.e. 5's) I end up memorizing of course, but things like 7's not so much. I suspect Randall is the same way. -boB (talk) 16:49, 3 June 2020 (UTC)
In fact, your example, where 7*8 screams 56 to you? I used my method and, like Randall, kept coming up with 54. 8*5=40, plus two more sevens (14) = 54. Feels right, but for some reason it's not. Took some head scratching to realize... it should have been two more EIGHTS, not two more sevens. Here's where the memorization thing would have been better. -boB (talk) 16:55, 3 June 2020 (UTC)

Well, 2,2 that's actually 2^3=8. 2,3 is addition instead of multiplication. 1,2 is division instead of multiplication. 1,1 is subtraction. 10,10 seems to be a visual gag, though most of the 10s row is either multiplication by 11 or 12... There's some logic to some of these, but it's different for each row, column, or cell. 21:13, 29 May 2020 (UTC)

Yeah, there is something going on. It looks like a lot of it is remembering the correct answer to a different problem. By my count 55 squares are the correct answer to a square next to it and 31 have a correct answer for somewhere else on the grid. Also, 2*2, 4*4 and 5*5 are double the correct answer. 21:41, 29 May 2020 (UTC)

It's almost disappointing that he didn't hide one or two asymmetries in there just to throw us off! 22:04, 29 May 2020 (UTC)

I think that was his nerd-snipe, there are no asymmetries (currently). 18:20, 2 June 2020 (UTC)

I get the idea that this is the sort of table you'd get if you tried to train an Adversarial AI from scratch to determine x*y purely by stochastic guessing and comparing to a co-evolving 'scorer' that also starts off naively but supports each answer according to the 'rightness' it thinks it has except for the real answer which is always hard-coded to be down-scored. (Also noting that DA reportedly came by his choice of 42 by asking people which numbers were 'funnier' than others, which can be said to be a similar kind of process but without the arrayed "original multiplication" element.) 22:13, 29 May 2020 (UTC)

As someone who often confuses 7*8 as 54, I found the alt text very humorous. 22:29, 29 May 2020 (UTC)

I'm disappointed to see that 6*9 isn't equal to 42. Probably not Douglas Hofstadter (talk) 23:01, 29 May 2020 (UTC)

This is just a collection of equations with the wrong answers. I'm not sure who finds this funny. 00:33, 30 May 2020 (UTC)

It's not funny per se, it's relatable.

https://www.explainxkcd.com/wiki/index.php/1210:_I%27m_So_RandomOverlord of oddities (talk) 01:16, 30 May 2020 (UTC)

I have asked a Code Golf Stack Exchange question with the goal of producing the shortest program that computes this function. Aaron Rotenberg (talk) 02:29, 30 May 2020 (UTC) @Aaron I had a similar thought, but was going to settle for the generator function for the main diagonal. If we can come up with one, we should submit it to https://oeis.org/ Cellocgw (talk) 13:30, 1 June 2020 (UTC)

I'm disappointed that 17 does not show up in any product cell, seeing as I've known since at least 1970 that 17 is the world's most random number. <-- a fact proved for a limited case here: https://www.youtube.com/watch?v=2JPSJL7Kvus Cellocgw (talk) 13:35, 1 June 2020 (UTC)

I too was unimpressed with this... unitl I got to the alt text. I'm in my sixties now, and for some reason, 8*7 has ALWAYS been difficult for me. I find myself always doube-checking to make sure I did it right. And 6*7 gave me problems too, but I got over that a few decades ago. I wonder what it is about those that gave us trouble. -- 14:35, 1 June 2020 (UTC)

I do not believe that the table was constructed rationally, but intuitively by Randall. He took the two factors (in both permutations) and thought, which resulting number he felt best about. It is more like a psychological experiment than a table constructed with a system or code in mind. Sebastian -- 16:27, 1 June 2020 (UTC)

Am I the only one who reads the author's intention slightly differently? I don't think that he intended that these values feel more correct than the real multiplication table. Rather, I thought he meant that from all possible wrong values, these feel most correct to him. In this way, I at least could sympathise with many values given here. 17:34, 1 June 2020 (UTC)

Agreed--he doesn't feel that the answers are incorrect, but rather, if he were given the problem on a test, "True or False: One times two equals one half.", he'd have to think for a minute. 18:08, 1 June 2020 (UTC)
I just edited that (first) bit, myself, before seeing your comments. I hope this version is better for you (might need further editing later in the paragraph, but stil considering this). 19:12, 1 June 2020 (UTC)

I don't know whether I am reading too much into this, but couldn't it be an allusion to all contemporary anti-science and anti-rational movements? We hear a lot of times from this kind of people that they do not need big professors to tell them what is true, because they now what "feels right" to them. What feels right to them may be just as wrong as this multiplication table, but that does not stop them to keep believing it. 21:28, 1 June 2020 (UTC)

"...this probably doesn't mean anything." Douglas Adams himself confirmed that the 6*9 in base 13 was a coincidence. He said himself in a BBC interview, "I don't make jokes in base thirteen." -- 02:22, 2 June 2020 (UTC)

"The title text (referencing Randall's suspicion that 6x7=42 may be wrong) is an allusion to The Hitchhiker's Guide to the Galaxy," Are we sure that this is an allusion to HGG? Not everything that has 42 is an allusion to HHG, there's no mention of the ultimate question or answer, nothing but a comment that 6x7 doesn't feel like it should be 42. Seems pretty irrelevant to include it in the explanation.Argis13 (talk) 23:22, 3 June 2020 (UTC)

Oh, haven't had a "Randy, get out of my head" moment for a while. After the rhyme and reason of 5x5 and 6x6 the multiplication by 7 has always felt off to me, too.

All of use who spent a few hours trying to find a formula have been nerd-sniped. 15:02, 4 June 2020 (UTC)

Hmmm. I'm sure these some logic to these numbers, we just can't see it. I 'feel' they're not random. Anyway, who can get their one times table wrong ? Also, is there anything solid about the text being a H2G2 reference ? Other than the mention of 42 ? Does H2G2 mention 56 ? And why do the whole multiplications of 6's, 7's and 8's (yeh I know, errant apostrophe's but it feels better this way) get people (including me) all at sixes and sevens (apparently worldwide - Chinese & Russians too). 12:21, 7 June 2020 (UTC)

___ I immediately interpreted this comic as a social commentary about the increasingly subjective ("feeling"-based approach) our world leaders and their influencees have to well established fact. This may not be the author's intention but it left me feeling satirically satisfied. 05:30, 9 June 2020 (UTC)

I STILL get 7x7 as 42 in my head sometimes. User:GetPunnedOn