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This sandbox is intended for saving drafts of long, complex answers, especially answers whose composition takes a long time. It serves to localize to one thread the front-page "bumps" caused by edits to drafts of such answers, so that they may be easily ignored. Also, it helps to guard against losing longly-composed answers due to system crashes.

To use this sandbox, look for a free answer below which says "This answer is free for anyone to use". If one exists then please use it. Else create a new answer. Start writing your answer there. Be sure to save a first draft quickly, to minimize the chance that that someone else concurrently tries to use the same free answer. Then, edit it as often as you like. When you reach a point that you'd like to post it to the intended target answer, simply copy it there and post it.

When your answer is complete and you no longer need the sandbox draft, please "clear" the draft by deleting all of the text and replacing it with the text "This answer is free for anyone to use". This is important because it eliminates TeX code (which may slow down MathJax rendering).

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I have added a [sandbox] tag to allow people ignore it more easily (via software support of ignoring tags), and since it seems that we have two sandboxes now, a tag may seem a bit more in place here. – Asaf Karagila Jul 18 '12 at 8:35
+1, great idea! It might be preferable if moderators delete "free for use" posts from time to time. You don't want two users writing their extensive, carefully crafted answers as edits of the same post and overwrite each other. Also, Chrome crashes from time to time. For some reason, I don't lose what I've written in an original post, but lose it when editing. So if I would use this thread, I would still prefer to create a new post. – Michael Greinecker Jul 18 '12 at 11:06
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@MichaelGreinecker, since deleted answers are still visible for 10k+ users, they would just accumulate in a useless state if they were deleted rather than reused. Also, if you prefer working in a pristine post rather than saving-and-then-editing (such as due to the autosave), then this sandbox is probably not for you anyway. You could do that just as easily at the eventual location of your post. – Henning Makholm Jul 18 '12 at 11:13
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(+1) For thinking outside the (sand)box. – cardinal Jul 18 '12 at 19:40
My own practice for sandboxes is to post a stub question and then delete it immediately. If you don't leave the page, you can continue to edit the deleted question, and then undelete it when you are finished. – MJD Jul 23 '12 at 18:14
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I have discovered a drawback of the method I described in the previous comment: moderators might choose to undelete your sandbox without consulting you. If you do use this method, it is probably a good idea to attach a header that says draft in large letters. – MJD Jul 24 '12 at 13:40
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I must say I am somewhat curious to why there are up/down-votes on the sandbox answers... – Willie Wong♦ Jul 24 '12 at 13:44
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@Mark Please don't delete any answers in this sandbox. Deleted answers are displayed to users with 10k+ rep. If every answer was deleted when done, then the thread would eventually have hundreds of deleted answers, which would make page navigation and MathJax rendering very complex for 10k+ users. That is why I advocated reuse of answers. – Gone Jul 24 '12 at 14:14
@Willie My guess would be that they are used to keep the answers sorted (so that they are not permuted randomly and user who edited one of the answers finds it in the same place when it returns). Perhaps it is slightly more comfortable that way. – Martin Sleziak Aug 6 '12 at 8:37
I find the sandbox is too slow to edit a long answer. – Makoto Kato Aug 26 '12 at 0:37
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@Makoto MathJax can get very slow when rendering long answers. I don't think the sandbox plays any role in what you see, since there is no other TeX code on the page. – Gone Aug 26 '12 at 2:15
@BillDubuque But it's a lot faster when I edit the same answer in the Ask Question box. – Makoto Kato Aug 26 '12 at 2:21
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@MakotoKato, unfortunately, you have chosen the worst possible browser to use. IE8 is slower than all other browsers (e.g., IE9 and IE7 are twice as fast, chrome is 5 times as fast for MathJax). Switching browsers would give you an immediate performance improvement. Short of that, the incremental preview that I link to above would help you a lot as well. I will look into it more, but IE8 has such poor performance, I doubt I can do much to help. Do you shut down your browser between uses, or is it running all the time? – Davide Cervone Aug 29 '12 at 10:33
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I just did some timing tests, and here are the results. Editing Makoto's post (with over 400 equations), to retypeset the entire page: in IE8, 2 minutes 25 seconds, in IE9, 12 seconds, in Safari 5, 10 seconds, in Firefox 15, 13 seconds. So you see that IE8 is really a bad choice or this type of work. The reason it is so bad is that any time MathJax asks for the size of some piece of mathematics (so it can place the superscripts properly, etc), IE reflows the entire page, and that slows things down considerably... – Davide Cervone Aug 29 '12 at 11:18
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At the suggestion of the moderators, I have gone and changed the associated owners of all the answers here to the Community user. This way, the original owners will not receive excess pings for each time another user uses the draft space for their work. Enjoy! – Grace Note♦ Oct 5 '12 at 14:45
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7 Answers

up vote 2 down vote

Spoiler test.

>! math.stackexchange is a wonderful place!

produces the spoiler below.

math.stackexchange is a wonderful place!

The spoiler should begin after an Enter key and the Enter key also acts as a delimiter.

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up vote 1 down vote

Let G be the set of ordered pairs $(a,b)$ where a and b are rational numbers and $a\not= 0$. Let $(a,b)*(c,d)=(ac, ad+b)$.

  1. Show that $G$ forms a group under $*$
  2. Show $A = \{(a,b) \in G: a=1\}$ forms a subgroup.
  3. Does $B=\{(a,b) \in G: b=1\}$ form a group? Explain your answer.
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up vote 1 down vote

As suggested by Adar, induction is a possible solution.

We rewrite the given inequality as

$$\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^n}\left(\frac{i}{n}\right)^{n-i}\left(\frac{n-i}{n}\right)^{i}\le\dfrac{1}{2^{n+1}}$$

The left-hand side of the inequality can be interpreted as follows.

We have $n$ boxes which are initially white. Place $n$ balls independently and randomly in the boxes; each box may contain more than one ball. Then, paint the $n$ boxes independently and randomly either blue or red. Say there are $i$ red boxes and $n-i$ blue boxes. Call a configuration of balls and boxes "good" if there are $n-i$ balls in the red boxes and the remaining $i$ in the blue boxes. Then the left-hand side is simply the probability that we obtain a "good" configuration.

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up vote 0 down vote

This answer is free for anyone to use

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up vote -1 down vote

This answer is free for anyone to use.

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up vote -1 down vote

This answer is free for anyone to use

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link – Goos Apr 17 at 16:52
up vote -2 down vote

This answer is free for anyone to use.

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