## Marsaglia’s Mental RNG

### January 29, 2019

Since this is a mental exercise, there is really no program to write. Here is the sequence of seeds, starting from 23:

> (define rs (do ((rs (list 23) (cons (+ (quotient (car rs) 10) (* (modulo (car rs) 10) 6)) rs))) ((member (car rs) (cdr rs)) (reverse (cdr rs)))))

> rs (23 20 2 12 13 19 55 35 33 21 8 48 52 17 43 22 14 25 32 15 31 9 54 29 56 41 10 1 6 36 39 57 47 46 40 4 24 26 38 51 11 7 42 16 37 45 34 27 44 28 50 5 30 3 18 49 58 53)

Here we compute the random digits by mapping the seed sequence mod 10, and compute a potentially-winning sequence for your next high-stakes rock-paper-scissors match:

> (define (mod n) (lambda (x) (modulo x n)))

> (map (mod 10) rs) (3 0 2 2 3 9 5 5 3 1 8 8 2 7 3 2 4 5 2 5 1 9 4 9 6 1 0 1 6 6 9 7 7 6 0 4 4 6 8 1 1 7 2 6 7 5 4 7 4 8 0 5 0 3 8 9 8 3)

> (map (lambda (n) (case n ((0) 'rock) ((1) 'paper) ((2) 'scissors))) (map (mod 3) rs)) (scissors scissors scissors rock paper paper paper scissors rock rock scissors rock paper scissors paper paper scissors paper scissors rock paper rock rock scissors scissors scissors paper paper rock rock rock rock scissors paper paper paper rock scissors scissors rock scissors paper rock paper paper rock paper rock scissors paper scissors scissors rock rock rock paper paper scissors)

As expected, there are nearly-equal counts of all digits:

> (uniq-c = (sort < (map (mod 10) rs))) ((0 . 5) (1 . 6) (2 . 6) (3 . 6) (4 . 6) (5 . 6) (6 . 6) (7 . 6) (8 . 6) (9 . 5))

But looking at adjacent pairs of digits show some bias:

> (define ps (let loop ((rs (append (map (mod 10) rs) (list ((mod 10) (car rs))))) (ps (list))) (if (null? (cdr rs)) (reverse ps) (loop (cdr rs) (cons (cons (car rs) (cadr rs)) ps)))))

> ps ((3 . 0) (0 . 2) (2 . 2) (2 . 3) (3 . 9) (9 . 5) (5 . 5) (5 . 3) (3 . 1) (1 . 8) (8 . 8) (8 . 2) (2 . 7) (7 . 3) (3 . 2) (2 . 4) (4 . 5) (5 . 2) (2 . 5) (5 . 1) (1 . 9) (9 . 4) (4 . 9) (9 . 6) (6 . 1) (1 . 0) (0 . 1) (1 . 6) (6 . 6) (6 . 9) (9 . 7) (7 . 7) (7 . 6) (6 . 0) (0 . 4) (4 . 4) (4 . 6) (6 . 8) (8 . 1) (1 . 1) (1 . 7) (7 . 2) (2 . 6) (6 . 7) (7 . 5) (5 . 4) (4 . 7) (7 . 4) (4 . 8) (8 . 0) (0 . 5) (5 . 0) (0 . 3) (3 . 8) (8 . 9) (9 . 8) (8 . 3) (3 . 3))

> (sort (lambda (a b) (or (< (car a) (car b)) (and (= (car a) (car b)) (< (cdr a) (cdr b))))) ps) ((0 . 1) (0 . 2) (0 . 3) (0 . 4) (0 . 5) (1 . 0) (1 . 1) (1 . 6) (1 . 7) (1 . 8) (1 . 9) (2 . 2) (2 . 3) (2 . 4) (2 . 5) (2 . 6) (2 . 7) (3 . 0) (3 . 1) (3 . 2) (3 . 3) (3 . 8) (3 . 9) (4 . 4) (4 . 5) (4 . 6) (4 . 7) (4 . 8) (4 . 9) (5 . 0) (5 . 1) (5 . 2) (5 . 3) (5 . 4) (5 . 5) (6 . 0) (6 . 1) (6 . 6) (6 . 7) (6 . 8) (6 . 9) (7 . 2) (7 . 3) (7 . 4) (7 . 5) (7 . 6) (7 . 7) (8 . 0) (8 . 1) (8 . 2) (8 . 3) (8 . 8) (8 . 9) (9 . 4) (9 . 5) (9 . 6) (9 . 7) (9 . 8))

Rearranging that so it makes a little bit more sense, we see:

0: 1 2 3 4 5 1: 6 7 8 9 0 1 2: 2 3 4 5 6 7 3: 8 9 0 1 2 3 4: 4 5 6 7 8 9 5: 0 1 2 3 4 5 6: 6 7 8 9 0 1 7: 2 3 4 5 6 7 8: 8 9 0 1 2 3 9: 4 5 6 7 8

The bias is evident in the ascending sequences of digits, some of which wrap around, although the ascending sequences aren’t ordered in the random sequence. Of course, there must be bias, since the calculations are so small there isn’t time for the generator to overcome the bias. But it’s still a worthy way to generate random digits in your head. If you want to know more, Marsaglia’s method is an instance of his multiply-with-carry random number generator.

You can run the program at https://ideone.com/Y9pNuZ.

I found the last digits were not distributed randomly. For a sequence of 10000 numbers the digits 1…8 appeared 1035 +- 2 times while 0 and 9 appeared ~862 times

Here’s a Python version as a generator:

I’m not sure it’s a particularly good PRNG, as each seed simply produces the same sequence in rotation, but it certainly is simple to use.

Note: DO NOT use 59 as the seed :)

Well, it’s a pretty good RNG considering it’s only got 6 bits of state! If the shortage of 0 and 9 is bothersome, you can always skip values greater than 50:

It’s interesting to generalize a bit and work in base b (this is a lag-1 Multiply with Carry RNG, as Praxis says). The inverse operation to bx + y -> ay + x is just n -> bn mod (ab-1) ie. the sequence, starting and ending at 1 is just the reverse of the sequence 1,b,b^2,b^3,… taking each element mod m = ab-1 so the period is the period of b, mod m. For a given b, this is longest when m is a prime and b is a primitive root mod m, and we do particularly well if we can take a = b-1 since the sequence then covers nearly all of the range [1..b*b]. For example, for b = 127, a = 126, we get a sequence of length 16000 including all but 129 of the 16129 two digit base-127 numbers. Here’s a program that calculates such maximal sequences:

Like 127, 999 is interesting (this is a CMWC generator, I believe):

I don’t think that sequence is in the OEIS.

Here’s a solution in C.

Example usage:

$ ./a.out 32 10

2

5

1

9

4

9

6

1

0

1