# Randomizing papers

I have a stack of paper documents that I would like to rearrange in a random order. They are standard letter (or A4) size office paper, so they're too large and flimsy to easily riffle shuffle like playing cards.

Each document may be a single sheet, or multiple sheets (up to 10 pages) attached with a staple in one corner. It is undesirable for multi-sheet documents to become separated during the process, as it will be difficult to reassemble them.

(A typical case is that the documents are exams submitted by students, which I need to grade. I do not have control over the order in which they are turned in, but I would like to randomize them to minimize the possibility of unintentional grading bias based on order of submission. My default approach is to pull papers arbitrarily from the middle and put them on top, but I don't think this results in a sufficiently random distribution.)

Suppose the number of documents is on the order of 100.

Bonus points for a method that results in a uniformly random permutation of the documents, or converges to the uniform distribution relatively quickly.

• As I am fairly new to this site, I'd appreciate if downvoters could give me a hint as to what problems they see with the question. I did read the guidance in the help center and did my best to follow it. May 2 '16 at 20:28
• I'd say it's being down voted as it doesn't seem to require a 'lifehack'. you should randomise the printing order rather than randomise the documents after printing.
May 3 '16 at 3:48
• @JustDoIt: If there's an "easy and readily available solution" then it's news to me, and I'd love to know what it is. I'm not rejecting arbitrary paper-pulling just to make the problem more interesting - it really isn't satisfactory. May 3 '16 at 15:10
• @JustDoIt: Because it doesn't solve my problem! May 3 '16 at 15:11
• The 'easy & ready-made solution' is to sort them alphabetically. Then there is no bias, no math & no mess. May 3 '16 at 16:30

Take a well shuffled deck of cards and paperclip a card to each one in turn. Then sort them into deck order A..K spades followed by A..K hearts, A..K diamonds, A..K clubs.

• So this reduces to the problem of sorting a pile of papers - but I presume that is "well studied". May 3 '16 at 14:36
• I don't understand what you mean by well studied.
– Dave
May 3 '16 at 16:22
• well studied means that you practically applied the concept of playing cards. Of course with deep thinking. May 4 '16 at 13:30

Sit down with your stack of documents and a D100 (gaming equipment, usually seen as a pair of D10, one designated as tens and the other as units). Roll the die, and count down the stack the number esimated, and transfer that document to a new stack. Reroll if there aren't enough documents for a given roll (optionally, switch to smaller dice as convenient, i.e. D20 when there are only 20 documents left). When there is only one document remaining, transfer it to the new stack.

This method should approximate a uniformly random condition after a single pass.

A potentially more time-effective variation on this method would be to set up a document rack, row of folders, etc., and sort the documents into the rack based on the die rolls, taking the top document from the original stack but putting it in the slot indicated by the roll. Re-rolling duplications would given the same measure of randomness in the final result, and progress would be significantly faster.

• Good plan, and I think "repeat three times" is redundant - if I'm not mistaken, you have a perfectly uniform distribution after just one pass (assuming fair dice). The only thing that worries me is that counting up to, say, the 53rd document in a stack of 100 is rather awkward and time consuming - and it makes this algorithm `O(n^2)`. May 2 '16 at 21:43
• @NateEldredge: Indeed, one pass is already totally random. To make it easier, I would lay all the documents out separately first. For example, lay them in a 10x10 grid and then use the two D10s. One die will give the column and the other the row of the document. If you only have D6's, then lay out 3 grids of 6x6. One die tells you which grid and the other two give the column and row. May 2 '16 at 22:55
• So basically waste 1-2 hours that could have been used in a more productive manner May 3 '16 at 16:12
• @JustDoIt: I think your comment about wasting time is unwarranted given the original question. The original question before edit did not include the use case regarding students and exams. I took him at his word that he actually needed a totally randomized stack of papers. I consider the use of dice to be a lifehack to this problem. I think the solution from Dave is even better and consider it to be more lifehack-ish. May 4 '16 at 11:23
• It's blowing an easy task out of proportion, so yes it will still end in a waste of time @James May 4 '16 at 15:23

When your students finish the test, have them roll a die and flip a coin. The result of the die determines which of six piles to place the test, and the coin determines whether the test goes on the top or bottom of the pile.

The ultimate goal is for the tests to be in a random enough order before you even touch them.

Messy but simple...

Throw them up in the air. Collect them up again.
Making sure they're all the right way round isn't really necessary, as you're going to go through them one at a time anyway.

Caveats:
Don't do it outdoors.
Don't do it near water.
Maybe check the stapling quality before a multi-sheet paper-shower.

The question was not to get even redistribution, merely randomisation.
The actual randomisation of this will work well, assuming sufficient ceiling height. The gym would be a good location - high ceiling, very little chance of losing any.

This may also provide some amusement for the students.

• With ten-page exams, this seems unnecessarily risky in terms of having sheets come loose from the stapled sets. May 3 '16 at 11:27
• Hence the caveat - but it seems a whole lot simpler than any method which requires actually counting, especially as the count is changed each time you remove a paper, leaving you with the dilemma, 'Do I put some kind of placeholder in where I just took the paper from, or do I reduce my count each time?' Without a placeholder, higher numbers are heavily favoured. May 3 '16 at 11:40

OK, as my first, rather frivolous method seems to not be convincing people, method 2 - still requiring no math, mental or otherwise & would be far more gentle on larger paper-sets...

...again, I am not going for perfectly random redistribution, just sufficiently disordered.

In any order you like, lay down sufficient sheets to fill the desk, in some semblance of a regular pattern.
Using your own judgement, repeat the process for layer 2.
Note that if you like, you can put a 3rd paper on one pile, or leave one off another pile. Continue until you are out of papers.
Pick up the piles, again using your own judgement of something approximating a random order.
Further randomisation can be achieved by picking up each pile with obverse or reverse face up.

Repeat the entire process, if necessary.

• Better than the "raining papers" method, anyway. Large chance of (unintentional) bias => no confidence of a genuinely random distribution, however. May 3 '16 at 12:14
• I've not been convinced yet that "absolute random" is even necessary. The order the students finish the paper is not a true indication of their ability in the first place. It will already have intermixed the 'just wanna get out of heres' with the 'smart but over-cautious' May 3 '16 at 12:20
• None the less, approaching a true statistical random is a desideratum in the question as it stands at present. This is to avoid "first paper" or "last paper" bias in the grading, I believe. Faster students may be seen as smarter by the teacher (or slower ones as more thorough); randomizing the paper order before grading and covering the names will help reduce or remove that potential for bias. May 3 '16 at 12:23
• Strongly agree with your comment @Tetsujin this question is a clear example of over complicating things. How fast or how slow a student is while answering a test is no factor of their knowledge or ability May 3 '16 at 16:14

I have done this.

Number the papers you wish to randomize in order, 1 to 100, say. If you want, you can turn the papers face-down to minimize the possibility of recognizing the paper and introducing bias inadvertently. Put the numbers on the back.

Get or make a table of random numbers from 1 to 100, say. There are some fine randomizers online that I have used. Just search for a number randomizer and enter the range you wish to generate.

Using the table of random numbers choose each numbered paper from the pile in turn according to the next number in the table.

Done.