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Sarah called me today and explained that she had counted to infinity. I shrugged and said it was impossible. She said that since I didn’t believe her, she would do it again, and this time in only ten minutes. I thought it was impossible but she did it right before my eyes!

How did Sarah count to infinity in only ten minutes?

Hints

Sarah started slow, but as time went on she got incredibly fast!

Clarifications

Sarah indeed counted all the way to infinity. She provided mathematical proof that she could count any infinite set, to include $א‎_0$, in any finite time span.

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    $\begingroup$ @MrPie All the way to infinity! $\endgroup$ – PerpetualJ Apr 29 at 23:40
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    $\begingroup$ This question feels a bit too broad. Are you sure there's one demonstrably correct answer to this one? $\endgroup$ – PiIsNot3 Apr 30 at 0:29
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    $\begingroup$ An appropriate question for your username, @PerpetualJ. $\endgroup$ – Gareth McCaughan Apr 30 at 0:59
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    $\begingroup$ My concern is that there may be more than one valid answer, and without specifying further restrictions, the “correct” one becomes an arbitrary choice. There’s already many good answers that could potentially be correct, which feeds into my concern $\endgroup$ – PiIsNot3 Apr 30 at 4:06
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    $\begingroup$ @PerpetualJ Several different answers already do so. As it stands, this seems to be too broad. $\endgroup$ – Deusovi Apr 30 at 10:49

17 Answers 17

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This feels underspecified:

clearly Sarah is not counting 1, 2, 3, ... (infinitely many numbers go here), infinity; so she's doing something else; but there are quite a lot of something-elses that she could do, and all of them are kinda cheaty, and the question here is what specific kinda-cheaty thing she did.

Here are a few possibilities. One:

She wrote numbers down on their sides, starting at 1 and proceeding as far as 8. An 8 on its side looks very much like the usual mathematical symbol for infinity.

Two:

She started from, let's say, "infinity minus 100" and counted up. (There are in fact number systems in which something a bit like "infinity minus 100" is an actual number.)

Three:

She counted down from, let's say, "infinity plus 100". (You can do something like that in the surreal numbers, mentioned above, but also in other simpler systems such as the ordinal numbers.)

Four:

She started counting normally, and at some point went "... and so on; infinity." I personally wouldn't (ahahaha) count that as counting to infinity, but then I don't think I'd count anything as counting to infinity other than the thing she obviously didn't do.

Five:

Sarah is able to count arbitrarily fast (maybe she's an archangel or something, not a human) and she said each number twice as quickly as its predecessor; after twice the time it took her to say "one", she had named all the positive integers and then said "infinity".

Apparently that last one is what the OP had in mind. Here are some more details.

Suppose it takes her two seconds to say "one", and then each new number is said 0.5% faster than the previous one -- so the next number takes 1.99 seconds, the next just over 1.98 seconds, etc. Then counting all the positive integers takes $2\left(1+\frac{199}{200}+\left(\frac{199}{200}\right)^2+\left(\frac{199}{200}\right)^3+\cdots\right)$ seconds, which equals $\frac2{1-\frac{199}{200}}$ or 400 seconds. This gives Sarah plenty of time to take a big breath and add "infinity", all within ten minutes.

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    $\begingroup$ I think you're missing an option: she counted to five -- she counted "All" "the" "way" "to" "infinity" as five words, which is a valid interpretation of what she said $\endgroup$ – postmortes Apr 30 at 6:31
  • $\begingroup$ @postmortes must be a very slow counter then if it takes her ten minutes ;) $\endgroup$ – Mark Apr 30 at 9:23
  • $\begingroup$ @Mark I agree :) $\endgroup$ – postmortes Apr 30 at 9:23
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    $\begingroup$ Gareth, your last one is the closest explanation! We all know there is no final number, but you can count an infinite set in a finite time span for sure! $\endgroup$ – PerpetualJ Apr 30 at 10:37
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    $\begingroup$ OK, done. I decided to make her speed up more slowly. 10 minutes is plenty of time :-). $\endgroup$ – Gareth McCaughan Apr 30 at 11:36
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I find it hard to believe she managed this in only 10 minutes, but all she needs to do is count to 1,461,559,270,678...

she just needs to do it in base 36, in which case the digits of the number are INFINITY.

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    $\begingroup$ She could count 200+ billion digits less if she used base 35 $\endgroup$ – b a Apr 30 at 7:33
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    $\begingroup$ Arguably this is the only answer yet that actually describes a finite process that could in any way be described correctly as "counting to infinity". (Arguably.) $\endgroup$ – Gareth McCaughan Apr 30 at 10:02
  • $\begingroup$ maybe she rot13(pbhagrq ol 7OW2OG (442628489 onfr gra) yrnivat bayl 3302 vgrengvbaf, rnfvyl qbar va 10 zvahgrf!). on a side note, who knew rot13(gur cevzr snpgbef bs VASVAVGL jrer (onfr 10) 2, 13, 127, 149, 18923, naq 2970661?) $\endgroup$ – SteveV Apr 30 at 17:08
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There goes one Infiniti G35! And there goes another! Infiniti G35 There. I've counted two Infiniti. ;)

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    $\begingroup$ Hahaha this was awesome lol $\endgroup$ – PerpetualJ Apr 30 at 2:19
  • $\begingroup$ Ah, the punch buggy no punch back answer :) $\endgroup$ – Captain Man Apr 30 at 14:56
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Perhaps

clever Sarah went the appropriate "I" page in the dictionary and counted word entries until she reached "infinity"

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    $\begingroup$ If so, she didn't count to infinity, she (at best) counted to "infinity". (Personally I don't think even that is a correct description of what she did.) But I don't expect whatever answer OP has in mind to be much more convincing than this. $\endgroup$ – Gareth McCaughan Apr 30 at 1:00
  • $\begingroup$ I don't see how this fits with the "Sarah started slow, but as time went on she got incredibly fast!" hint. $\endgroup$ – Simon Baars Apr 30 at 11:12
  • $\begingroup$ It was posted before that hint was added. (As were almost all the answers, including one of mine that already does the slow-then-faster thing. I suppose the hint was added more to make the question less too-broad than because it was needed as a hint :-).) $\endgroup$ – Gareth McCaughan Apr 30 at 11:37
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    $\begingroup$ But that's only to "infinity" - far too low. If you instead listed Disney catch-phrases, then you could count "To Infinity, and Beyond" $\endgroup$ – Chronocidal Apr 30 at 14:09
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Did Sarah count

All of the avengers movies up to and including infinity war?

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She counted "to infinity", 10 letters, 1 space.

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    $\begingroup$ Ah, that explains the timing as well, unlike some of the other answers! (+1) Good first post! :P $\endgroup$ – Mr Pie Apr 30 at 5:34
  • $\begingroup$ That explains ten minutes? $\endgroup$ – Peregrine Rook Apr 30 at 22:26
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How about counting:

$\frac{1}{1000}, \frac{1}{999}, \frac{1}{998}, \cdots, \frac{1}{2}, \frac{1}{1}, \frac{1}{0}$

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    $\begingroup$ Your final term is not defined... but.... meh ;) $\endgroup$ – Mr Pie Apr 30 at 4:48
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I can do it in 15 minutes. Duh.

enter image description here

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  • $\begingroup$ Do you actually mean 40 minutes? $\endgroup$ – Arnaud Mortier Apr 30 at 9:28
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    $\begingroup$ @ArnaudMortier 15 minutes because he rotated it 90 degrees (desperately trying not to spoil) $\endgroup$ – Steve-O Apr 30 at 13:09
  • $\begingroup$ @Steve-O Thanks, I see it now. $\endgroup$ – Arnaud Mortier Apr 30 at 13:10
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Possible mathematical answer

I think this is linked to

Zeno's paradoxes

Possible approach

Sarah defines for you a new number system. The number $1$ is represent by saying the letter "a" for a duration of $10$ seconds, the number $2$ is represented by saying the letter "a" for $5$ seconds, the number $3$ is represented by saying the letter "a" for a duration of $2.5$ seconds and, in general, the number $n$ is represented by saying the letter "a" for a duration of $\frac{10}{2^{n-1}}$ seconds.

She then says "a" for a duration of $20$ seconds to count to infinity.

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  • $\begingroup$ There’s a name for this phenomenon! $\endgroup$ – PerpetualJ Apr 30 at 10:54
  • $\begingroup$ @PerpetualJ I've added an extra line. Is this the phenomenon you're talking about? $\endgroup$ – hexomino Apr 30 at 10:57
  • $\begingroup$ Surely that is counting to 0, not to ∞? (since n = 1+ln(10/t)/ln(2)) $\endgroup$ – Chronocidal Apr 30 at 13:30
  • $\begingroup$ The 20 seconds, is counting 1,2,3,4,5,... consecutively. I understand there is ambiguity here (1 is the same as 2 2s etc) but I decided it was not important to address in the context of the problem. $\endgroup$ – hexomino Apr 30 at 13:41
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The definition of infinity in some circles is

the highest conceivable number.

Therefore, all Sarah needs to do is count to

the highest number she knows of, be that a hundred, a thousand, whatever. Because she cannot think of any number higher than that, that is her "infinity".

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    $\begingroup$ Finitism much (it seems that Sarah is accepting the highest number she can count to as infinity)? $\endgroup$ – MilkyWay90 Apr 30 at 1:17
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Sarah is also known as

Chuck Norris

Indeed:

"Chuck Norris counted to infinity. Twice."

And since

"Chuck Norris has his own Gender.", Sarah is a suitable second name for them.

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Sarah isn't great at counting but she is great at improving anything she does while she is doing it.

Therefore, every time she counts a number she can count it faster than the previous one. The improvement it's not that spectacular, and to count one number still takes her 99.8% of the time it took to count the previous one. This way, if counting to 1 took Sarah 1 second, the time it will take Sarah to count to n is: $1+1\cdot 0.998 + 1\cdot 0.998^2 + .... + 1\cdot 0.998^n$ Since that's just a geometric series, it's sum to infinite is $\frac{1}{1-0.998}=500$ seconds. That is, just thanks to keeping improving continuously, Sarah can count to infinite in 8 minutes and 20 seconds.

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    $\begingroup$ I like this one! I actually didn’t expect another answer to come in with good mathematical arguments! +1 $\endgroup$ – PerpetualJ Apr 30 at 10:45
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The answer is:

She starts counting and for each number she takes half the time to count to the next number, through this method you can count to infinity in a finite time.

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Despite the puzzle being already solved, I have another take on this.

All she needs to do is

use a diverging function.

For instance, she could say

-log(3), -log(2), -log(1), -log(0)

Which is in agreement with the hint that she goes incredibly fast in the end.

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Could this be the right approach (even if it is not the correct answer)?

Time is finite, yes, but continuous, so it contains an infinite number of individual positions. If we apply, for example, the function f: x -> 1/(10-x) to the interval of minutes [0,10] belonging to the Real Numbers, just before the 10 minutes we will have reached infinity.

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Did Sarah perhaps say once:

$\lim\limits_{x\to\infty} x = \infty$

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  • $\begingroup$ Sarah would have to say this very slowly to reach ten minutes, which would be impractical if @PerpetualJ was paying for every minute during the phone call! $\endgroup$ – Mr Pie Apr 30 at 8:23
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My answer is:

As Sarah got really fast, time stopped...according to Einstein's relativity..hence she did it....I know this sounds absurd but ...meh!

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