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## APA Formatting and Citation (7th Ed.) | Generator, Template, Examples

Published on November 6, 2020 by Raimo Streefkerk . Revised on August 23, 2022.

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## Raimo Streefkerk

## Combinatorics

Select 1 unique numbers from 1 to 100, features of this random picker.

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## Pen and paper pseudo-random number generator

How can I generate pseudo-random numbers using only pen and paper?

- Uniform distribution (or as close as possible)
- It's pen writing on paper; can't cut, fold, throw or anything like that
- I'm expecting some formula or algorithm, but a geometric solution, graph or set theory would be ok, too
- It's really just pen and paper. No clocks or anything external

math.stackexchange already has some questions on this, like this and this .

So, the definition of a pseudo-random number generator in the context of this question becomes:

A formula or algorithm, that, given an initial seed, generates a non-cyclic sequence of digits. The distribution of the digits in the sequence should be balanced.

I have uploaded to gist the Notebook I used to play with the proposed solutions:

https://gist.github.com/caxcaxcoatl/3d469a9728737e51530add73327f9c9f

- $\begingroup$ Can a deck of cards be used? If so you could use the solitaire cypher as a key stream source. $\endgroup$ – Q the Platypus Sep 14, 2019 at 4:44
- $\begingroup$ @Q the Platypus, no just pen and paper... nothing external $\endgroup$ – caxcaxcoatl Sep 14, 2019 at 5:07

## 3 Answers 3

- $\begingroup$ That's exactly what I was looking for. After reading the Wikipedia article, I experimented a bit and ended up with m=97 , a=13 and c=0 (though I have no idea what a primitive element is, or if 13 can be considered as such in this context). This produces a cycle of length 96, almost perfectly balanced for mod 2, 5 and 10. For mod 2, the cumulative sum of the frequencies of 0 and 1 got a somewhat big difference between items ~30 and ~90 (for a seed of 7), but I think this would be too convoluted a process for mod 2, anyway. $\endgroup$ – caxcaxcoatl Sep 15, 2019 at 3:35
- $\begingroup$ just one thing, I think the way you wrote the formula in your response is incorrect. The way it is, I read the modulus operation happening on c alone, and not on aXn + c $\endgroup$ – caxcaxcoatl Sep 15, 2019 at 3:40
- 1 $\begingroup$ A primitive element $a \bmod m$ is one that the smallest $k$ such that $a^k\equiv 1 \pmod m$ is $k=m-1$ If $m$ is prime there will be a number of primitive elements. They generate the multiplicative group $\bmod m$, so all numbers from $1$ to $m-1$ can be represented as powers of $a$. An example that is primitive is $3 \bmod 7$ because the powers of $3$ are $3,2,6,4,5,1$, while $2 \bmod 7$ is not primitive because the powers of $2$ are $2,4,1$ and do not include $3,5,6$ $\endgroup$ – Ross Millikan Sep 15, 2019 at 4:41

## Square roots

In that case, the square root should be an irrational number, which is acyclic by definition.

I'm not accepting this as the answer for my own question because:

- The analysis above shows the distribution can be quite unbalanced
- Calculating the square root by hand is somewhat expensive; a simpler algorithm would be welcome
- I'm just generally looking for other options

- $\begingroup$ Very interesting. Algorithmic and looks very simple. I just need some clarification: when you refer to the last answered question , are you actually referring to the last item of the seed? Looking at the Wikipedia article, it looks like it does not require any input other than the initial seed. Or is the actual last answered question input into it somehow? $\endgroup$ – caxcaxcoatl Sep 15, 2019 at 23:27
- $\begingroup$ The last answered question. So it will be the eighth question at first, then as you fill in the ninth question it becomes the last question at the place you start counting from. $\endgroup$ – Q the Platypus Sep 15, 2019 at 23:36
- $\begingroup$ @Q, Ok, I think I got it. So, I fill the first eight with some values out of my head, and then the rest of the test with the register. As I would have already answered some random questions in the test, I found it easier to use the register disconnected from the order of the questions: I start with a seed, say, TFFTFTTF, then I apply the algorithm whenever I want to fill a question with a random value (say, question 37, if I know I won't be able to answer it, then 29, if that's the next one I need) $\endgroup$ – caxcaxcoatl Sep 16, 2019 at 3:41

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Research Randomizer is a free resource for researchers and students in need of a quick way to generate random numbers or assign participants to experimental conditions. This site can be used for a variety of purposes, including psychology experiments, medical trials, and survey research.

Have you tried the new Fullscreen Wrapping Random Number Generator ? Watch as the wrapping paper is ripped off layer by layer to reveal a random number. <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-M83SJW5" height="0" width="0" style="display:none;visibility:hidden"></iframe>

Generate numbers sorted in ascending order or unsorted. Separate numbers by space, comma, new line or no-space. Download the numbers or copy them to clipboard Click on Start to engage the random number spinner.

The random number generators above assume that the numbers generated are independent of each other, and will be evenly spread across the whole range of possible values. A random number generator, like the ones above, is a device that can generate one or many random numbers within a defined scope.

Given a seed X n it returns X n + 1 = a X n + c ( mod m) for parameters a, c, m. I would then take some low order bits of X n + 1 as the random number. The calculation is not much even with pen and paper. You could use a relatively small m because you don't need many bits.