# A needle in a Pi-stack

Recently I’ve been watching a lot of find-in-pi source code

(require racket/generator)

(define (make-pi-spigot)
(generator ()
(let loop ([q 1] [r 0] [t 1] [k 1] [n 3] [l 3])
(if (< (- (+ (* 4 q) r) t) (* n t))
(begin
(yield n)
(loop (* 10 q)
(* 10 (- r (* n t)))
t
k
(- (quotient (* 10 (+ (* 3 q) r)) t) (* 10 n))
l))
(loop (* q k)
(* (+ (* 2 q) r) l)
(* t l)
(+ k 1)
(quotient (+ (* q (+ (* 7 k) 2)) (* r l)) (* t l))
(+ l 2))))))


Simple enough to use, we can use Racket’s for to generate a list of n digits of pi or to convert the same to a string:

(define (digits-of-pi n)
(for/list ([i (in-range (+ n 1))]
[d (in-producer (make-pi-spigot) #f)])
d))

(define (pi->string n)
(string-append
"3."
(apply
string-append
(map number->string (cdr (digits-of-pi n))))))

> (digits-of-pi 10)
'(3 1 4 1 5 9 2 6 5 3 5)

> (pi->string 10)
"3.1415926535"


In any case, it seems to be working well enough so far, so let’s write some code that can look for any arbitrary list of digits in pi:

(define (find-in-pi ls)
(let ([pi (make-pi-spigot)])
(pi) ; skip the 3
(let loop ([i 1] [l* '()])
(let* ([d (pi)]
[l* (map cdr (filter (lambda (ea)
(and (not (null? ea))
(= d (car ea))))
(cons ls l*)))])
(if (any? null? l*) (- i (length ls)) (loop (+ i 1) l*))))))


Using this, we can find the first time each digit occurs in pi:

> (for/list ([i (in-range 10)]) (list i (find-in-pi (list i))))
'((0 31) (1 0) (2 5) (3 8) (4 1) (5 3) (6 6) (7 12) (8 10) (9 4))


Or the first time that longer sequences occur:

> (find-in-pi '(1 2 3))
1923

> (find-in-pi '(4 5 6))
250

> (find-in-pi '(0 0 0))
600


Now, let’s get crazy and see when Jenny's number shows up:

┌ ☺ [email protected] ~/Projects/SmallProjects
└ time ./find-in-pi 8675309

8675309 is 9202591 digits into pi


I added a few more lines to the full source that lets it run from the command line and compiled it with raco exe find-in-pi.rkt.

That took a while to run. 😄