I just read this interesting xkcd strip:

On a scale of 1 to 10, how likely is it that this question is using binary?

At first I thought it was funny, but as I got to ruminate a little over it, I was surprised to be unable to find an answer. As Karolis Juodelė pointed out, the probability is ε, as there is an infinite number of bases containing 1 and 10.

However, to get a finite answer, we can modify the puzzle like this:

On a scale of 1 to 10, how likely is it that this question is using binary vs. decimal?

So my question is: How should I solve this puzzle? Is there a correct answer at all? Is this what we call a self-reference paradox, like Multiple-choice question about the probability of a random answer to itself being correct?

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    This is not a puzzle. There is no method of evaluating the likelihood of things happening in other peoples heads. If you just want to get the right base, say $1$. – Karolis Juodelė Jan 11 '14 at 09:54
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    Actually, if you assume that all possible interpretations of $1$ and $10$ are equally likely, the probability is $0$, because all bases have $1$ and $10$. This could be base $2$, base $10$ or base $123456$. – Karolis Juodelė Jan 11 '14 at 09:57
  • Base $10$, definitely base $10$, [all number bases are base $10$](http://math.stackexchange.com/questions/166869/is-10-a-magical-number-or-i-am-missing-something) – peterwhy Jan 11 '14 at 10:28
  • @KarolisJuodelė, I meant the probability of binary vs. decimal. I edited my question for clarity now. – osvein Jan 11 '14 at 14:50
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    She doesn't know what a 4 is, so it is either binary, ternary or base-4. (It can't be unary because 10 is a number). – Empy2 Jan 11 '14 at 15:24
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    @anustart in your modified version, if your question is in binary (the word in common sense), then I don't understand your "2" there, but still your question is in base 10. Alternatively, if your question is in, say, tridecimal (the word in common sense), then your question is not in base 2 but in base 10. So I will definitely vote for base 10, like I said above. – peterwhy Jan 11 '14 at 18:37
  • @peterwhy Oh, I see your point now, that base-10 is not the same as decimal if the question is base-2. – osvein Jan 11 '14 at 20:29
  • AFAIS if the question is decimal, the answer is `1+(10-1)\2 = 5.5`, but if it's binary, the answer is `1_2+(10_2-1_2)\10_2 = 1+(2-1)\2 = 1.5`. – osvein Jan 11 '14 at 20:48
  • This type of question is not to be taken seriously. But if I read the strip correctly, there is the indication that $4$ is not a known digit, so it is definitely not in decimal. Taking as (completely unreasonable) axiom that digits must be taken from the list $0,1,2,3,\ldots,9,A,B,\ldots$ in that order, the base can only be two, three, or four. – Marc van Leeuwen Jan 12 '14 at 08:40
  • @MarcvanLeeuwen It is not a puzzle, I am asking whether there is a single correct answer that does not depend on itself. If there is an answer that will be the same no matter whether the question is binary or decimal. I didn't ask this question to "riddle" others. – osvein Jan 12 '14 at 09:55
  • @anustart: My point is that one should take into account the fact that the strip mentions the digit $4$ (twice), and something should be done with that information, which none of the comments/answer so far do (except one). Since the digit appears to cause puzzlement (my interpretation of the drawing), I conclude that it is **definitely not decimal**. And asking about (only) binary vs. decimal is certainly not implied by the strip, other bases must be considered too. – Marc van Leeuwen Jan 12 '14 at 11:05
  • @MarcvanLeeuwen I'm looking for a generic answer, an answer that is correct no matter what happens in the asker's head. – osvein Jan 12 '14 at 13:28
  • @anustart: MU.. – Marc van Leeuwen Jan 12 '14 at 13:32

1 Answers1


(If I understand correctly, $10$ means certainly in binary and $1$ means certainly in decimal)

To answer your modified question, if I were to think like in your last comment (2014-01-11 20:48:42Z), then I would answer:

As likely as $1+\dfrac{10-1}{1+1} = \dfrac{11}{1+1}$

But for me, I choose to answer

As likely as $1+1+\dfrac{1+1-10}{(1+1)^{1+1+1}}$

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  • If the question is decimal, the scale is 1 to 10, which means 1 = never binary and 10 = always binary, and also 5.5 will be a 50/50 chance. If the question is binary, the scale is 1 to 2, which means 2 = always binary and 1.5 (or 1.1 as binary) will be a 50/50 chance. Therefore I'd choose to answer "If decimal 5.5, if binary 1.1". Is there a single answer that will be correct independent on its base? – osvein Jan 12 '14 at 07:15
  • How about `(b+1)/2` where `b` is the base? – osvein Jan 12 '14 at 07:23
  • That is equivalent to the first answer I gave above: $\frac{11}{1+1}$. – peterwhy Jan 12 '14 at 07:25
  • Ok, then I just realized the if the question is binary, the answer is binary as well. +1! – osvein Jan 12 '14 at 07:28
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    But then, why assign 50:50 chance to 1 and 10? If the question is in decimal, then 1, if binary then 10. – peterwhy Jan 12 '14 at 07:37
  • Ok, I get it now. You're absolutely right. Thanks! – osvein Jan 12 '14 at 10:01