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This is quite a soft question but hopefully, this community has some nice answers.

What is a good way to "toss a coin" in your head? That is an "algorithm" to generate heads $50/50$ of the time roughly that is hard for you to influence without thinking too deeply.

I am looking for little mathematical tricks to "randomise" and scramble ones ability to select a handful or manageable numbers. In a way that is hard to forsee, predict or influence like the listed below. I am not looking for ways to manipulate real-world data like freckles, hair, second hands, books or digits of pi.

My methods so far

Mod 3:
I think of the first two digit numbers that come into my head, multiply them together and if the result is $1$ mod $3$ I say heads, if $2$ mod $3$ I say tails and if $0$ mod $3$ I go again. This seems to work quite well but now I know how the game works it is quite hard to not overinfluence my choices without picking very large numbers.

Collatz down
I think of a number quickly and then perform the collatz operations on it, halfing if even and tripling then adding one if odd. If it takes an odd number of steps to reach $1$ I say heads and if even I say tails. I worry this doesn't yield heads $50\%$ of the time however.

Penultimate digit
I think of two $2$ digit numbers and multiply them together. If the tens digit is even I say heads, if it is odd I say tails.

Can someone think of other quick methods that are hard to influence?

OskarSzarowicz
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  • Quite interesting question actually! I like it ... – Matti P. Oct 13 '21 at 09:27
  • Humans are bad "random generators". Try to forget your previous choices , then you might produce somehing more or less random. – Peter Oct 13 '21 at 09:27
  • Likely relevant: https://stackoverflow.com/questions/3919597/is-there-a-pseudo-random-number-generator-simple-enough-to-do-in-your-head – Matti P. Oct 13 '21 at 09:28
  • @MattiP. thank you for that link, glad you liked the question! – OskarSzarowicz Oct 13 '21 at 09:29
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    I think sometimes poker players look at the second hand on their watch to generate a random number. – littleO Oct 13 '21 at 09:31
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    Perhaps the algorithm selection can depend on how fast and how many times you want to perform this. Perhaps a reasonable first approach would be to think/look for a word and see if it has odd or even number of letters. But that's something that's hard to repeat many times and keep it consistent. – Matti P. Oct 13 '21 at 09:32
  • @littleO I thought of lots of ways to do this if you had pins, watches, rubber bands or pencils but I am looking for ways to do this in your head! thank you though – OskarSzarowicz Oct 13 '21 at 09:32
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    You need to find sources of randomness first. Every "generator" of randomness needs something random to start with, such as the time , or "starting with the first numbers that come to my head" which has some initial distribution, or like the answer below, tracking apples and yoghurt (or articles in the newspaper). Which is the best tracker is up to opinion. My personal opinion is that you should phrase this question in such a way that you already have with you a presupposed source of randomness such as a watch or a newspaper, and from this , you want to randomly, and efficiently ,choose. – Sarvesh Ravichandran Iyer Oct 13 '21 at 11:30
  • However, there is a more interesting principle involved here : I haven't quite seen it anywhere and expect it to be true, but if you begin with an initial source of very little randomness, then you can't increase the randomness too much if you insist on doing too little algorithmic work as well. This means that you either need to do a lot of algorithmic work, OR you need to start with a very amazing source of randomness. I'm not sure the latter question has any definitive mathematical answer, but the first question and the principle are more interesting. – Sarvesh Ravichandran Iyer Oct 13 '21 at 11:46
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    See https://math.stackexchange.com/questions/2094304/how-to-mentally-flip-a-coin and https://math.stackexchange.com/questions/2485498/mentally-generating-a-pseudorandom-0-1-sequence-with-uniform-distribution – leonbloy Oct 13 '21 at 11:52

2 Answers2

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Think of the amount of the most random things and then add them together, then mod 2.

For example: Number of apples I saw today + Tubs of youghurt I have in the fridge.

Or:

Think of a random word and find the middle letter(s). Then find its number value, A = 1, B = 2, ... , Z = 26. If that's too hard, If the letter is in the first half of the alphabet, etc.

If the middle number is 2 letters, pick another word

Aaa Lol_dude
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  • I really like your second method! It will be hard to remember all the letters for quite a long time! – OskarSzarowicz Oct 13 '21 at 09:36
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    Here's a even better idea - if the letter belongs to the first half of the alphabet or not. – Aaa Lol_dude Oct 13 '21 at 09:38
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    I think that is slightly worse actually because it is easier to influence! In a sense that if I can quickly control heads or tails because I know letters like E, B and D are quite early on and letters like O, R and X are quite late. Also the density of letters I would assume things like X,Y,Z are much less common than A,B,C – OskarSzarowicz Oct 13 '21 at 09:42
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    After checking the letter frequency the first half of the alphabet is bigger than the second half at around 0.1%.

    Then again, you can use either.

     – Aaa Lol_dude Oct 13 '21 at 09:44
  • Second method is good, but as all vowels give an odd number, I guess the chance for an odd number is significantly higher than for an even number. Better use that first half-second half approach, which is easier to apply anyway. The separation between first and second half could be adjusted such that the chance are about the same. As you write in your last comment that is not needed, so it'll be A--M and N--Z. – azimut Oct 13 '21 at 11:17
  • Yes, it should be it. – Aaa Lol_dude Oct 13 '21 at 11:37
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This suggestion is based on the frequency of occurrence of letters in the English Language.

It is estimated, from frequency analysis, that 49.89% of letters are from the list below (they are not in frequency order but chosen to make the correct total and make it easy to remember):

O R N A T E.

If you want to include Q as well, that makes it 50%

Other selections are also possible, of course.

Therefore, have an English book or newspaper in front of you, close your eyes and pick a letter at random.

If the chosen letter is on this list, then it's heads, otherwise it's tails.

David Quinn
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    This feels a bit like cheating for me. Using a book as an external source of entropy, it is not entirely "in your head" as the OP had asked for. – azimut Oct 13 '21 at 12:14
  • I realy like this method David! I think my question is ill posed. I want to find a way to "scramble" numbers in a way that is hard to predict or influence. Your out of the box thinking is really cool however – OskarSzarowicz Oct 13 '21 at 13:35
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    Thanks @OskarSzarowicz . I was trying to think of a method that didn't involve arithmetical calculations, but it seems some people think it is cheating to use an external source.. Oh well, never mind, eh? – David Quinn Oct 13 '21 at 17:27
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    @DavidQuinn you cant please them all! – OskarSzarowicz Oct 13 '21 at 18:19