# euler 26 reciprocal cycles

Feb 11, 2015

Problem: find the number less than 1000 with the largest repeating cycle of digits in the decimal expansion of its reciprocal

As always, the initial reacion was on the lines of aha! this is obvious!:

``````(defun get-repeated-substring-length (str len)
(let* ((rev-str (reverse str))
(offset (- (length str) (* 2 len)))
(match (equal (subseq rev-str 0 len)
(reverse (subseq str offset (+ offset len))))))
(if match
len
0)))

(defun get-cycle-length (dec)
(declare (type double-float dec))
(let* ((dec-str (princ-to-string dec))
(exp-str (subseq dec-str
(1+ (position #\. dec-str))
(position #\d dec-str))))
(do ((i 0 (1+ i))
(cycle-length 0 (get-repeated-substring-length exp-str i)))
((or (> cycle-length 0) (> i (/ (length exp-str) 2))) cycle-length))))

(defun inverse-cycle-digits (n)
(let ((inverse (/ 1.0 (coerce n 'double-float))))
(do* ((i 0 (1+ i))
(dec (* 1.0 inverse) (* 10.0 dec))
(cycle-length 0 (get-cycle-length dec)))
((> cycle-length 0) cycle-length))))

(defun euler-26-fail ()
(let ((all-cycles (mapcar #'inverse-cycle-digits (loop for i from 1 to 1000 collect i))))
(reduce #'max all-cycles)))
``````

But the aptly named `euler-26-fail` fails to do what we want, because once again, the largest reciprocal when expanded fully, lies beyond the standard double-float range. So, we hand-roll …

``````
(defun inverse (n &key (precision))
(do* ((i 0 (1+ i))
(numerator 10 (* 10 (mod numerator n)))
(dec (floor (/ 10 n)) (floor (/ numerator n)))
(declist (list dec) (cons dec declist)))
((> i precision) (nreverse declist))))

(defun cycle-length-helper (declist)
(do ((i 0 (1+ i))
(cycle-length 0 (get-repeated-substring-length declist i)))
((or (> cycle-length 0) (> i (/ (length declist) 2))) cycle-length)))

(defun cycle-length (declist)
(loop for length in (loop for i from 1 to (length declist)
collect (cycle-length-helper (subseq declist 0 i)))
maximizing length))

(defun get-all-inverse-lengths (lst max-precision)
(mapcar #'(lambda (n) (cycle-length (inverse n :precision max-precision)))
lst))

(defun unique-max (list)
(let ((max (reduce #'max list)))
(if (= 1 (count max list))
max
0)))

(defun biggest-cycle (max-num prec)
(let* ((lst (loop for i from 1 to  max-num collect i))
(lengths (get-all-inverse-lengths lst prec))
(max-len (reduce #'max lengths)))
;;(format t "Debug: lengths = ~A, max = ~A~%" lengths max-len)
(nth (position max-len lengths) lst)))
``````

The idea is to loop over all numbers with a given precision, find the max, then maybe increase the precision and try again. This is a terrible idea, relying on the ability to submit multiple answers at the Project Euler website :(

But more than that, it’s terribly slow. It becomes marginally faster if we restrict our search to primes.

``````(defun biggest-cycle (max-num prec)
(let* ((lst (eratosthenes max-num))
(lengths (get-all-inverse-lengths lst prec))
(max-len (reduce #'max lengths)))
;;(format t "Debug: lengths = ~A, max = ~A~%" lengths max-len)
(nth (position max-len lengths) lst)))

(defun sieve (prime list)
(if (null list)
(list prime)
(cons prime (sieve (first list)
(remove-if #'(lambda (n)
(= 0 (mod n prime)))
(rest list))))))

(defun eratosthenes (max-num)
(let ((all-numbers (loop for i from 2 to max-num collect i)))
(sieve (first all-numbers) (rest all-numbers))))
``````

With this change, `(biggest-cycle 1000 1000)` yielded `499` (in 211 seconds), but it turned out to be incorrect. `(biggest-cycle 1000 10000)` took 22810 seconds, which is shameful, but yielded the right answer.

### Update

Here is an effort to redeem myself:

``````import qualified Data.List as L
import qualified Data.Vector as V

eratosthenes :: Int -> [Int]
-- ^Get a list of prime numbers less than the given number.
eratosthenes n =
let x = [2 .. n] in

sieve :: Int -> [Int] -> [Int]
sieve p nums =
let rest = filter (\x -> x `mod` p > 0) nums
in if null rest
then [p]
else p : sieve (head rest) (tail rest)

inverse :: Int -> Int -> V.Vector Int
-- ^Given a number and the required precision, calculate the decimal expansion of its reciprocal.
inverse n prec =
let truncate x y = (x `div` y, x `mod` y)
nextDigit x count
| count == 0 = []
| otherwise = let (d,m) = truncate (x * 10) n
in d : nextDigit m (count - 1)
in
V.fromList \$ nextDigit 1 prec

getInverses :: Int -> Int -> [V.Vector Int]
-- ^Get a list of all the inverses of all numbers upto a given number, for a given precision
getInverses maxNum maxPrec = [inverse x maxPrec | x <- [1 .. maxNum]]

checkRepeatLength :: V.Vector Int -> Int -> Int -> Int
checkRepeatLength expansion end len =
let s1 = V.slice (end - len) len expansion
s2 = V.slice (end - len - len) len expansion
c = V.zipWith (==) s1 s2
in
if V.and c
then len
else 0

checkCycles :: V.Vector Int -> Int -> Int
-- ^Check if a cycle exists at a given end point with a given length; if it does, returns the length itself, otherwise 0
checkCycles expansion end =
let l = quot end 2
cycles = filter (> 0) \$ map (checkRepeatLength expansion end) [1 .. l]
in
case cycles of
x:xs -> x
otherwise -> 0

findCycleLength :: V.Vector Int -> Int
-- ^Get the length of a cycle at a certain end point; if none found, returns 0
findCycleLength expansion =
let l = V.length expansion
minEnd = quot l 2
cycles = filter (> 0) \$ map (checkCycles expansion) [l, l-1 .. minEnd]
in
case cycles of
x:xs -> x
otherwise -> 0

euler26 :: Int -> Int
-- ^Attempts to solve Euler Problem #26 for a given precision
euler26 prec =
let nums = [2 .. 1000]
cycles = map (\x -> findCycleLength \$ inverse x prec) nums
maxPos = L.elemIndex (maximum cycles) cycles
in
case maxPos of
Just n -> nums !! n
_ -> 0
``````

This gives the same answer (`euler26 10000`) in `7.5 seconds` (which is a 3000x speedup). Just to make sure you don’t jump to that conclusion, it isn’t a Haskell vs Lisp thing, it’s just calculating less stuff, and doing it with vectors instead of lists.

In particular, the first version was running for 12 hours before I killed it out of disgust, and realized that instead of checking if the `length` of the list was greater than 0, I should do a `case` match instead since I only needed the first element. This made a big difference. Also, this is the first time I didn’t have to rely on `Debug.Trace`, so I feel good about that :)