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Alice In Wonderland - "Base"-ic Math

Here's a tricky one. On about the third page of chapter two, Alice starts speaking out some bizarre equations...

"Let me see: four times five is twelve, and four times six is thirteen, and four times seven is -- oh dear! I shall never get to twenty at that rate!"

Alice is correct. 4*5=12... when expressed in base 18. Bases are different standards by which numbers are measured. We count in base 10, computers count in base 2 (a.k.a. binary). HTML colors are in base 16 (a.k.a. hexadecimal, like 1F(b16) is 31(b10)). So anyway, following this pattern...

4*5=12 (b18)

4*6=13 (b21)

4*7=14 (b24)

4*8=15 (b27)

4*9=16 (b30)

4*10=17 (b33)

4*11=18 (b36)

4*12=19 (b39)

4*13=1A (b42) (or about 32(b10) short of 20). The equation falls apart here. Alice will never get to 20 at this rate. :)

Special Requirements:

good knowledge of math

Avg. Rating:	7.1 of 10 - (1818 votes cast)
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Contributed By:

eric-at-bibliopile on 12-05-2000

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Comments:

Keith writes:: This will be added to my list of "The Most Intelligent Things I Have Read In All My Years On The Web". My list now contains one (1) item.; 249 of 272 people found this comment helpful. Did you?

ComputerMan writes:: Base number conversion (from base 10) is accomplished by dividing a number by the target base. The least significant digit (in the result) is the remainder of the first divide operation. Alice starts with 4 x 5 = 20(b 10). To convert 20 to base 18, divide by 18, (i.e., 20 / 18 = (1 r. 2)). You repeat this operation using the quotient as the next dividend to get the next digit. You stop when the quotient is zero (i.e., 1 / 18 = (0 r. 1)). So, the first remainder is 2 and the second is 1, yielding the result of 12. DETOUR: To see how this works with a larger number, lets convert 749 (b 10) to base 18. 749 / 18 = (41 r. 11); 41 / 18 = (2 r. 5); 2 / 18 = (0 r. 2). Now we use the remainders in reverse order: 2, 5, then 11, but we cannot use the 11 in that form. We need a single symbol to represent that value. As the example from eric-at-bibliopile suggests, the convention is to use uppercase alphabetic characters (A, B, C, D…) to get the necessary number of symbols. So, we use A B to represent 10, B to represent 11, etc. thus, 749 (b 10) = 25B (b 18). Now, back to Alice: When you convert 24(b10) to base 21: 24 / 21 = (1 r. 3) yielding 13. Note: 20(b N) is 2 * N(b 10), so 20(b 18) = 36(b 10) and 20(b 21) = 42(b10). In Alice’s problem, N (the target base) is determined at each step by adding 1 to n (the multiplier), then multiplying by 3. (i.e., N = ((n+1)x3)). Since the product of (4 x n) is increasing by 4 and the base number is increasing by 3, the step-by-step results get farther away from 20(b N) at each step. That is to say, the difference between 20(b N) and the product (4 x n) increases at each step (i.e. 36–20=16; 42–24=18; 48–28=20; 54-32=22; etc.) Indeed, Alice will never get to 20. Also note: If you extrapolate backwards, you discover that the pattern starts at 4 x 3 = 10(b12) followed by 4 x 4 = 11(b15), while smaller numbers don’t fit: 4 x 2 = 8(b 9); 4 x 1 = 4(b 6).; 97 of 109 people found this comment helpful. Did you?

Matt writes:: The author, C.L. Dodgson (writing under the pseudonym of Lewis Carroll), was also a mathematician / logician, for those who were not aware. He wrote many articles and books on mathematics and logic. He was also a mathematics lecturer at Christ Church College in Oxford, appointed in 1855. The things you learn in a discrete math class...; 48 of 57 people found this comment helpful. Did you?

The Birthday Princess writes:: Mathematician Martin Gardner credits the base egg info above to Alexander L. Taylor's 1952 book "The White Knight." Gardner also offers in his book "The Annotated Alice" another reason Alice can't reach twenty. Traditionally times tables go to the twelves. Alice can only reach 4 x 12 = 19 (according to her logic) and not beyond to 4 x 13 = 20.; 28 of 39 people found this comment helpful. Did you?

8-bit kid writes:: True- this egg is probably right. I still think the author grows pot in his basement, though..... "There are only 10 types of people in this world- those who can read binary and those who cannont"; 22 of 37 people found this comment helpful. Did you?

FifthStageErikson writes:: Actually, if anyone has ever read The Annotated Alice, they'll realize the volume of those type of eggs. It's like every other line has political or mathematical significance. Some of it is really awesome... Like *strains to remember* some of the comments about geography and gravity and things... So many little references! It's awesome.; 7 of 9 people found this comment helpful. Did you?

mamlam writes:: I find that 4 out of 5 people leaves 1 people. About the only thing in here that I understand.; 7 of 12 people found this comment helpful. Did you?

Scott Madden writes:: Ah, the Charles Dodgson side of the author strikes again!; 2 of 4 people found this comment helpful. Did you?

Dark Jabberwocky writes:: My god-you are BRILLIANT! How in the world did you come to that, were you just sitting around in a chair and said "wait a minute..."? Well now I wonder why in the world doesn't she use decimal(b10); 6 of 12 people found this comment helpful. Did you?

together we fight writes:: Dang you're good! Look, I kinda need a tutor for "Algebra 2" next year...; 11 of 22 people found this comment helpful. Did you?

DizzyElk writes:: I understood the explanation on the first read, but, then again, I got off lucky. I'm an IT worker, and have to learn decimal to binary or hexadecimal conversions.; 2 of 4 people found this comment helpful. Did you?

wormer writes:: TapanM, that "person" was queen Victoria. :-); 3 of 7 people found this comment helpful. Did you?

turtle writes:: This is the best egg i've seen. The easiest way to understand bases that I've heard of is those 1s, 10s, 100s blocks or sheets (shading) that people do when learning math.; 0 of 1 people found this comment helpful. Did you?

efren writes:: cool dude!!!; 21 of 44 people found this comment helpful. Did you?

angela_uk writes:: lol keith :); 4 of 10 people found this comment helpful. Did you?

Holly writes:: Wow, how did you EVER work that out?? I would never have noticed something like that!!!!; 6 of 14 people found this comment helpful. Did you?

sevendeadlysins writes:: I would just like to say that this shows massive intelligence!!! Good on ya!; 3 of 8 people found this comment helpful. Did you?

Alechko writes:: The book "The Annotated Alice" includes a note about both of these counting methods. The publisher of an egg probably just read the book. Also there are many more interesting facts about Alice in the book.; 3 of 8 people found this comment helpful. Did you?

Mayy writes:: That's incredible! I've been reading these easter eggs on this site for about an hour now, and that's the best one I've seen so far. Awesome! =]; 3 of 8 people found this comment helpful. Did you?

DrunkNmunky writes:: OH MY GOODNESS..... I got an instant headache when I saw that! HA HA, anyway, yeah, once I got rid of it though, and read that about fifteen times it actually makes some sense to me now...thats very cool.; 2 of 6 people found this comment helpful. Did you?

Bam-bam writes:: I was wondering why she was so bad at math! When I was reading the book, I was so confused! Thank you.; 1 of 4 people found this comment helpful. Did you?

fats writes:: Damn. And I thought REGULAR math was hard!; 5 of 12 people found this comment helpful. Did you?

mayor_of_loserville writes:: Seriously, "GP" writing in the present tense is just not that hard. Your grammar destroys your credibility.; 0 of 2 people found this comment helpful. Did you?

Natie writes:: i didn't get the comment at first. i was like, what on earth are you talking about???!?! but i read it soooo MANY times afterward and i also saw ComputerMan's comment and i get it now. that's truly awesome!!! math isn't all that bad....; 1 of 5 people found this comment helpful. Did you?

Pookey writes:: Groovy, Dude!!! I've alway's sort of wondered about that.; 1 of 6 people found this comment helpful. Did you?

sexynemesis writes:: Lewis Carroll was in fact an opium user, but his books were not based on this drug use. Lewis told these stories to a young girl (which by the way he was in love with) names Alice Liddle and her little sisters. In the book he contradicts mathematical definitions, like when Alice is running with the Red Queen, "We must keep running to stay in the same spot." he defies reality and tries to conceive what it would be like on the other side of the looking glass. Lewis Carroll was also a suspect in the Jack-the-Ripper murder case, reason being that supposedly his poem "the Jabberwocky" contained hidden anagrams, I have found this false, because "Alice" was written many years before the murders took place. the only thing Lewis was guilty of was being a pedophile, he was not schizophrenic either, but he did smoke opium, as did everyone else at the time, which, then it was considered harmless.; 16 of 37 people found this comment helpful. Did you?

GP writes:: Forget my previous egg, I will explain here. Let me explain why the MAIN calculation is WRONG. You can not calculate an with different BASE.. you should stay in BASE 18. The trick is she use the result of base 10 4x5=20 and the result of it decomposed in base 18 4x5=12 If you whis to add 3 for every base you need concret clue. Just because there is 3 between B18 and B21 is not enough to suggest that the next one will be B24 If you use this you can say anything you want. she say: four times five is twelve, and four times six is thirteen, and four times seven is if you stay on base 18 (we have to asume that 20 is in base 18) 4x5=12 4x6=13 then 4x7=14 ... 4x20=80 Is That is why she say it it be long before she will get 20. SHE PLAY WITH the calculation . she calculate 4x5 in base 10 and take the result and decomposite in base 18. 4x5 = 20 will be 12 because 20 is "1" x18 and "2" x1 ("12") 4x6 = 24 will be 13 because 24 is "1" x18 and "3" x1 ("13") That means 4x7 = 28 will be 14 because 28 is "1"x18 and "4" x1 ("14") So if we answer her question to reach 20 she should say: 4x20 = 80 will be 48 because 20 is "4"x18 and "8"x1 ("48") So her answer when she will reach 4x20 will be 48 And the big joke is here 4x13 = 52 in both base 10 and base 18.. since 4x13=20 if you add always 1 to the number multiplyed; 4 of 13 people found this comment helpful. Did you?

GP writes:: I think you are all wrong. If she use BASE 18, you must stay in BASE 18 She will reach 20 only when she will say 4x80=20 .. This is why she found it is far.. Or maby she is just making a joke about SHE WILL NEVER UNDERSTAND 4x13=20 because in base 18 4x13 = 52 in both BASE 18 and BASE 10 ..; 0 of 6 people found this comment helpful. Did you?

Lindsey writes:: Very interesting!; 4 of 16 people found this comment helpful. Did you?

GP writes:: my previous try was wrong. What about this one: 4x5 = 12 will be 12 because 7+5 = ("12") 4x6 = 13 will be 13 because 7+6 =("13") 4x7 = 14 will be 14 because 7+7 =("14") 4x8 = 15 will be 15 because 7+8 =("15") 4x9 = 16 will be 16 because 7+9 =("16") 4x10 = 17 will be 17 because 7+10 =("17") 4x11 = 18 will be 18 because 7+11 =("18") 4x12 = 19 will be 19 because 7+12 =("19") 4x13 = 20 will be 20 because 7+13 =("20") It is strange because this way I get 4x13 = 20 positively Maybe this way with the same result of 4x5 = 12 and 4x6 = 13 and incrementing by one we can obtain 20 quicker.. I got 20 on 4x13 exactly where there say it break appart. But to complet we should know what is "4x" , let say that "4 = 7" and "x = +" so let me continue 4x4 = 11 will be 11 because 7+4 =("11") 4x3 = 10 will be 10 because 7+3 =("10") 4x2 = 9 will be 9 because 7+2 =("9") 4x1 = 8 will be 8 because 7+1 =("8") 4x0 = 7 will be 7 because 7+0 =("7") So when She say she can not get 20 with this "RATE" it might means we should not use the "x" = Rate but x="Addition" with 4 being simply a fixed variable = to 7 proven because 4+0 = 7.; 1 of 12 people found this comment helpful. Did you?