# SMBC gets the finger

By Phil Plait | June 21, 2010 10:24 am

I’ve not seen this trick before, but Zach Weiner at Saturday Morning Breakfast Cereal is correct. Not only is he correct, but his math is correct, and his philosophical punch line is funny and correct. And I’m not saying that just because I’ll see him at Comic Con soon and I’m trying to get him to buy the first round. I swear, sometimes when I don’t get a math trick, that guy in the panel really is me.

Also. Don’t forget to hold your mouse over the red button at the bottom for extra bonus Zach-ish goodness*.

* He really does look like that.

CATEGORIZED UNDER: Cool stuff, Geekery, Humor
MORE ABOUT: math, SMBC, Zach Weiner

1. DaveH

That nagging “why?” he neatly chronicles strikes me as psychologically similar to the question “Why is there something not nothing?” and, generally, resistance to the anthropic principle.

Why? Because if it didn’t work it wouldn’t be a maths trick.

2. For some reason, my first test of the trick was on 5 * 5.

It was… unsatisfying.

3. Gus Snarp

It works because we have a base ten number system because we have ten fingers.

But I looked at this three different ways and though it was all wrong or didn’t work in all cases until I finally realized I was doing it wrong.

Ha! Cool, and the math is correct. Too bad it takes more time to do multiplication this way than to, you know, simply memorize the multiplication table

5. John Sandlin

I didn’t have the “but… why?” response of the lower panels. I wish I could say I figured it out before scrolling down, but I didn’t. It’s interesting that someone figured it out. The part that always amazes me is the creative inspiration that drives this type of discovery. Once explained it may seem so obvious, yet it took someone brilliant to notice and figure it out the first time.

6. Ben

Huh, neat, the only one of these I knew was the 9 * X version. Hold all 10 digits up and lower the digit at position X (counting from the left). The number of fingers to the left of the down digit is the tens place, the number of fingers to the right is the ones place. Never knew there was a version for more of the multiplication table.

7. Jo

GIANT RED BUTTON! And all this time I’ve been wondering why SMBC didn’t have easter eggs. I do fail. Now, must resist urge to go through the entire back catalogue…

8. AliCali

The website says it’s Polish. I learned this from my Persian parents when I was a kid (1970s), so I always thought it was a Persian thing. Maybe it’s a European thing.

It may take too long now, but when I was young, I got 6×7, 6×8, and 7×8 screwed up all the time. The finger-thing saved me.

9. Teresa

Hey Phil–
Darn you, darn you to heck! Now I have to go through the SMBC back catalog and look at all the red button easter eggs. This is worse than when I found out about the XKCD hover texts…..grumble…grumble….

10. XPT

Very neat, I didn’t know about it. However maybe it’s my Calculus classes, but the Why? question didn’t come up: perfectly satisfied with the proof.
Although isn’t it all because we happen to count in base ten and have 10 fingers?

11. MoonShark

I’ve heard of a couple variants on this trick. This is just the first time I’ve seen it done using fingers. Not sure exactly what to call them, but poke around on the web and I’m sure some will turn up.

Here’s one of my favorite math “tricks”. Post the next line if you figure out the pattern! If someone gets it right, then post the line after theirs if you figure it out too.
1
11
21
1211
111221

Extra credit: What’s the greatest single digit used in this pattern and why? (I actually don’t remember why)

12. WardVB

nice trick, will use it in my math classes!
and fun riddle:
312211

13. Idlewilde

my mind is stuck on a fibbonacci type series when i see that…

14. Mchl

Being Polish I can only say SMBC was the first place I’ve ever heard of this trick.

(Although there’s one book I might want to check, whe it could possibly be)

[edited]

Ok. I have it:
A book called ‘Lilavati’ by Szczepan Jeleński says this trick was described in ‘Khelasat as hissab’ (Of nature of calculation) by 16th century Syrian named Beha-Eddin.

The ‘Lilavati’ book (first published in 1926) itself is full of methematical puzzles and curiosities, and maybe ‘Elegant Polish lady’ learned the trick from it

@MoonShark: 312211

15. Gus Snarp

13112221

Phil’s evil twin Richard just did this. In my mind it’s run length encoding, but there are several names for the pattern.

16. jcm

He even provides a proof.

17. XPT

1113213211

@Moonshark.

SPOILER BEGIN!

Greatest digit is 3 as you can’t have the same 4 digits in a row. Numbers are made of couples, and you can’t have two couples such as (1,1),(1,1) which then would bring (4,1). 1111 could describe 11; but 11 brings 21, not 1111!

SPOILER END!

18. Jamey

I’d like to see a couple of more examples, as I’m not sure I am following how to get the finger patterns.

19. RPJ

I honestly didn’t get it. I got to the middle-why part, and was more like “Huh, neato. I wonder what the proof is.” which I suppose can be simplified to “why”.

When I looked at the proof and attempted to simplify and and got

10((x-5) + (y-5)) 1((10-x)(10-y))
———————– + ———————
10((x-5) + (y-5)) 1((10-x)(10-y))

=1 + 1.

Not really sure what I’m doing wrong there, unless I misunderstood what is meant by up values and down values. For instance, taking the example of 7 x 8, the “up” value of the left hand (2) can be found by subtracting a palm from it (x-5) and the “up” value of the right hand is found the same way (y-5 = 3; y=8; “up” value = 3). The multiplication expression works the same way, except by subtracting the original value from two palms instead. So, simply substituting numbers for words, I get the above expression.

Either way, “why” what? Why is stuff real? Why does math exist? why do you always get the same answer from the same question? These things seem to be self-evident.

20. SalsaShark

Last Friday I received my degree in mathematics, and now, on Monday, I’m learning new facts about grade 2 arithmetic.

And moonshark, the next number in the sequence is 1, and I’ll prove it:
x(n)=649419 – (7310448/5)n + (3523055/3)n^2 – (12887995/30)n^3 + (217363/3)n^4 – (136037/30)n^5

(Don’t believe me? Try it out with n=1, n=2, …, n=6 and you’ll see! Unless I made an embarrassing arithmetic error)

And to answer your bonus question, the greatest single digit used in this sequence or pattern is, well, 9.

I realise it isn’t customary to write questions on tests… but what the hay: how come I came up with a completely different answer than the one you were expecting, yet both my answer and your answer appear to be correct?

Mathematics scepticism ftw.

21. Hakan

RPJ, Zach is not doing a division, those are explanations. The proof is one line.

10((x-5) + (y-5)) + ((10-x)(10-y)) = 10 x + 10 y – 10 x – 10 y + xy = xy

so it works.

It turns out Zach’s parentheses suck but his awesome coolness compensates for it.

22. MoonShark

Good job folks, especially XPT

23. RPJ

@ Hakan: Ah, I get it now. Makes much more sense. Though I had constants in the expression:

10((x-5) + (y-5)) + 1((10-x)(10-y))

10x-50 + 10y-50 + (10-x)(10-y)

10x – 50 + 10y – 50 + 100 – 10y – 10x + xy

xy

Not sure where those keep disappearing to for everybody else.

24. Steve

As I use an HP calculator, I prefer the “reverse polish hand magic”.

25. Tim G

Don’t forget the trick done to Fingerman in Stand and Deliver

u is fingers up
d is fingers down

9*x = 10u + d (eqn 1)
u + d = 9 (eqn 2)

Manipulating (eqn 1) and (eqn 2) gives

u = x – 1

and

d = 10 – x

26. Pete

I don’t get it. How does it work for numbers under 5?

I completely fail to apply the trick to 3*4.

27. SalsaShark

It doesn’t work for numbers under 5 because (x-5) and (y-5), which are the number of up fingers on your right and left hand, respectively, would be negative numbers for x,y<5. It would work, if you somehow had negative fingers. Likewise, (10-x) and (10-5) represent the down values of your left and right hands, respectively. And you would need more than five fingers to put down on either hand in the cases where x,y 5 in those cases). It’s a biological limitation I suppose, not a mathematical limitation.

28. pumpkinpie

6×7=42
1+2 fingers up = 3
4×3 fingers down = 12

?

29. Lucy Kemnitzer

It doesn’t seem to work for 5 times 8:

0 fingers up + 3 fingers up equals 3

0fingers up times 3 fingers up equals 0

so 5 times 8 would get you 30, but it ought to get you 40. Or do you call the palm 1? No, that doesn’t work either, you would get 48.

Am I missing a step?

30. E

@pumpkinpie

(3*10) + 12 = 42

31. SalsaShark

3 tens, 12 ones, 3*10+12=42=6*7

32. Jeffersonian

My first try was 5×6.
Doesn’t work.
Neither did a few others.
Am I missing something? Is the joke on gullible people?
I suppose you could assign numbers to various body parts and then do math but what would be the point?

33. piesquared

@30 – read the method better. Here’s the example given, then your “exception”:

7×8 – 2 fingers up (3 down) on left hand, 3 fingers up (2 down) on right hand.

up: 2 + 3 = 5
down: 3*2 = 6
total: 5*10+6 = 56 = 7*8

5×6 – 0 fingers up (5 down) on left hand, 1 fingers up (4 down) on right hand.

up: 0+1 = 1
down: 5*4 = 20
total: 1 * 10 + 20 = 30 = 5*6

It works with every set two numbers (assuming you have enough fingers and can do negative fingers, or if you restrict yourself to 5-10 x 5-10), and the mathmatical proof is given in the comic.

34. Lucy Kemnitzer

Ok so: 0+3=3
3 time 10 =30
5 times 2 =10
30+10=40.

The thing I was missing was the fingers _ down_ in the second part!

Can the Polish aspect be confirmed, or only the Persian?

35. Poodle_Slayer

@Moonshark/SalsaShark

While run-length encoding is an elegant answer, I managed to find another pattern consistent with what you put up (provided I’m not sleep deprived and hallucinating). I had hackjob rule (albeit a little less hackjob than SalsaShark’s): Each digit ‘generates’ a 1 to its left in the next step, unless it is part of a ’11′ sequence in which case the ’11′ becomes a ’2′ in the next step and ‘generates’ a 1 to the right. In this case, the greatest digit would just be ’2′ unless I introduce new rules. Well, I guess SalsaShark made a point; there are many possible rules behind the pattern, especially if you have MATLAB.

36. Mchl

Lucy, see my comment #14.

37. Grizzly

Chisanbop anyone?

38. SalsaShark

@Poodle_Slayer

MATLAB ftw!

39. rob

you’re all wrong.

obviously the next line is:

42

40. RobT

Pretty neat! But…

7/10 for not showing the complete solution and skipping steps!

That’s the only problem I have with the explanation; if I did that in math class my teacher would have deducted marks. It took me a minute to figure it out since he skipped over a line or two. Like, what happened to the -5′s on the left and the 10′s on the right? I know, now, they cancel each other out but it isn’t so apparent at first.

Rob.

41. archgoon

@35
Given the first n numbers for a sequence, you can fit an (n-1) polynomial to them. This is almost always wrong.

42. Branespace

Now that’s a sequence I haven’t seen in a while. That would be the Cuckoo’s Egg, from “The Cuckoo’s Egg” by Clifford Stoll? Also known as look and say?

1
11
21
1211
111221
312211
13112221
1113213211
31131211131221

It’s a fun little sequence. The single greatest digit (within the nominal sequence, assuming starting with a 1) is 3. Why is that the case? I can’t really do a full mathematical proof for that kind of thing (Way way undergrad), but the gist is that you can have at most 3 of a number in a row. For instance taking the hypothetical seeded value:

1110

Reading it to the next line gives us 3 ones and 1 zero. Every time we have three of any number in a row, it collapses to a single or double, but because the count is, we would get 3110. This leads to 132110, and so forth. I hope this is clearer than I think it is.

Of course, you can always seed the first value as any number, which would allow us to experience a wider number of digits.

You can find a (much) better analysis in:
J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16

43. SalsaShark

@41 – what do you mean it’s almost always wrong? It’s a provable theorem. Do you mean it’s almost always not the right sequence? If that’s the case, than I beg to differ. That’s the problem with these “find the pattern” questions, as there is literally an uncountably infinite number of patterns which fit a given n terms. For ANY term in the (n+1)th spot there exists some algebraic pattern which fits the given n terms and the (n+1)th term. And that’s only algebraic patterns…. The only criteria someone has to say that the pattern someone comes up with is “wrong” is that the person trying to figure out the pattern wasn’t able to delve in the question-giver’s mind and find out which pattern he was referring to. Moreover, I think these questions are very harmful to math students as they attempt to teach pattern recognition, but because literally any number can be next and some pattern fits it, they are really only being taught how to best guess what the person asking the question is thinking.

44. dexmach

@SalsaShark

Well done on your alternative formula, and your latest post. You said more or less exactly what I was going to say. I hate pattern recognition, especially when included in intelligence tests… they’re not a measure of how clever you are, just how conformist your thinking is.

45. pumpkinpie24

@E
“@pumpkinpie

(3*10) + 12 = 42″

Right, I get that from the proof, but in the initial explanation it isn’t clear you have to multiply the first by 10 and add the 2nd. So it’s not really as simple as it is portrayed! Another step of math. Sheesh.

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