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Here's a challenge : find the shortest way to write the number $1234567890$ .

There is several ways to write the number $1234567890$ :

  • $1.23456789 × 10^9$
  • $2×3^2×5×3607×3803$
  • $617283945×2$

But all these notations are longer. Can you find a shorter notation than $1234567890$ ?

EDIT : For this question, the length of a notation is given by the number of characters used to write the notation on a sheet of paper.

eg : $2×3^2×5×3607×3803$ is 16 chars long.

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    $\begingroup$ @SanathDevalapurkar forum? Stack Exchange is not a forum. $\endgroup$ – John Dvorak Apr 9 '14 at 16:52
  • $\begingroup$ @SanathDevalapurkar Why ? Questions, problems, enigmas, aren't they challenges for the math-lover ? $\endgroup$ – Pyrofoux Apr 9 '14 at 16:54
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    $\begingroup$ $x$ in $1234567891$-numeral system where $x$ is used to denote $1234567890$ $\endgroup$ – drhab Apr 9 '14 at 16:56
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    $\begingroup$ In base 1234567890, it is written as 10 $\endgroup$ – Nick Matteo Apr 9 '14 at 16:56
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    $\begingroup$ You have to define a way to determine the length of a notation. See for example @Alraxite's answer: do you count the width of the graphical representation or the character used to type it in $\LaTeX$ or MathML code, or...? You get the idea. $\endgroup$ – rubik Apr 9 '14 at 17:17
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$$\quad\quad\quad$$




$$\tiny{1234567890}$$

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  • $\begingroup$ The length is measured by the amount of characters to write it on a sheet of paper, so this is no better than the original. I still found it funny, though. $\endgroup$ – Eric Apr 9 '14 at 18:26
  • $\begingroup$ Excellently funny and clever ! $\endgroup$ – Pyrofoux Apr 9 '14 at 18:32
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I stated this in a comment, so I might as well put it here. In the commonly accepted base64 notation (http://en.wikipedia.org/wiki/Base64), $1234567890 = BJlgLS_{64}$. 8 characters.

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How about: $\displaystyle\sum_{i=1}^{9}i\;10^{10-i}$


Alternatively, how about $123\cdots90$? It's only $8$ characters long!

(or the $7$ character $12\cdots90$ if you find the pattern unambiguous enough)

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  • $\begingroup$ Interesting answer, but it looks like it takes more space than $1234567890$. $\endgroup$ – Pyrofoux Apr 9 '14 at 17:56
  • $\begingroup$ @Pyrofoux I've added another suggestion. Does that work? :-) $\endgroup$ – Alraxite Apr 9 '14 at 18:21
  • $\begingroup$ I love it ! But where is the $0$ ? :p $\endgroup$ – Pyrofoux Apr 9 '14 at 18:31
  • $\begingroup$ @Pyrofoux Fixed. Thanks. $\endgroup$ – Alraxite Apr 9 '14 at 18:46
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It should be $$ 10 $$ in base $1234567890$ of course.

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What about $KF12OI_{36}$? If it were possible you could go up to base $99$, but as far as I know it's defined only for bases up until $36$.

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  • $\begingroup$ Base 64 is used very frequently $\endgroup$ – Davis Yoshida Apr 9 '14 at 17:22
  • $\begingroup$ @DavisYoshida: I know that for base $36$ alphanumerics characters are used. What do you use for base $64$? Don't you mean the base 64 encoding? $\endgroup$ – rubik Apr 9 '14 at 17:23
  • $\begingroup$ @DavisYoshida how do you denote $55_{64}$ by one digit -- rubik's notation is only good for base 36 (10 digits + 26 letters)? $\endgroup$ – gt6989b Apr 9 '14 at 17:23
  • $\begingroup$ @rubik The standard is A-Z,a-z,0-9,+/ $\endgroup$ – Davis Yoshida Apr 9 '14 at 17:24
  • $\begingroup$ Here is a link: en.wikipedia.org/wiki/Base64 $\endgroup$ – Davis Yoshida Apr 9 '14 at 17:24
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In base 32 it is $14PC0MI_{32}$ (9 digits together with the base, 1 shorter than the original)

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I don't know if this is acceptable, I've been working on an algorithm to find the numerical equivalence of any sequence given a dictionary.

This is the java code for this: https://github.com/volkovasystems/convert-to-sequence.git

Now by feeding this the following inputs:

  1. the sequence index: 1234567890
  2. the dictionary: abcdefghijklmnopqrstuvwxyz

You can get the sequence equivalence of 1234567890 from the given dictionary which is: "jvqowyc"

You can even make it shorter or longer depending on the given dictionary.

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