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Problem 28¶

Number Spiral Diagonals¶

Starting with the number $1$ and moving to the right in a clockwise direction a $5$ by $5$ spiral is formed as follows:

It can be verified that the sum of the numbers on the diagonals is $101$.

What is the sum of the numbers on the diagonals in a $1001$ by $1001$ spiral formed in the same way?

螺旋状に並んだ数の対角線¶

1から初めて右方向に進み時計回りに数字を増やしていき, 5×5の螺旋が以下のように生成される:

両対角線上の数字の合計は101であることが確かめられる.

1001×1001の螺旋を同じ方法で生成したとき, 対角線上の数字の和はいくつか?

(import (scheme base)
        (gauche base)
        (gauche array))

(define (diagonal-positions xynums)
  (assume (exact-integer? xynums))
  (assume (positive? xynums))
  (let* (; 左上→右下
         [diagonal-pos1 (fold-right
                          (^[n lis] (cons `(,n . ,n) lis))
                          '()
                          (iota xynums))]
         ; 右上→左下
         [diagonal-pos2 (fold-right
                          (^[n lis] (cons `(,n . ,(- xynums 1 n)) lis))
                          '()
                          (iota xynums))]
         [diagonal-pos (append diagonal-pos1 diagonal-pos2)])
    (if (odd? xynums)
      ; 奇数方陣の時は中央が重複して列挙されるので1つを外して返す
      (cons `(,(div xynums 2) . ,(div xynums 2))
            (remove (^c (= (car c) (cdr c) (div xynums 2)))
                    diagonal-pos))
      diagonal-pos)))

(define (diagonal-nums xynums)
  (assume (exact-integer? xynums))
  (assume (positive? xynums))
  (let ([table (make-table xynums)])
    (map (^c (array-ref table (car c) (cdr c)))
         (diagonal-positions xynums))))

(define (make-table xynums)
  (assume (exact-integer? xynums))
  (assume (positive? xynums))
  (letrec ([walk-right! (^[n y x table]
                          (cond
                            [(and (< (+ x 1) xynums)
                                  (not (array-ref table y (+ x 1))))
                             (array-set! table y (+ x 1) n)
                             (walk-down! (+ n 1) y (+ x 1) table)]
                            [(and (<= 0 (- y 1))
                                  (not (array-ref table (- y 1) x)))
                             (array-set! table (- y 1) x n)
                             (walk-right! (+ n 1) (- y 1) x table)]
                            [else table]))]
           [walk-down! (^[n y x table]
                         (cond
                           [(and (< (+ y 1) xynums)
                                 (not (array-ref table (+ y 1) x)))
                            (array-set! table (+ y 1) x n)
                            (walk-left! (+ n 1) (+ y 1) x table)]
                           [(and (< (+ x 1) xynums)
                                 (not (array-ref table y (+ x 1))))
                            (array-set! table y (+ x 1) n)
                            (walk-down! (+ n 1) y (+ x 1) table)]
                           [else table]))]
           [walk-left! (^[n y x table]
                         (cond
                           [(and (<= 0 (- x 1))
                                 (not (array-ref table y (- x 1))))
                            (array-set! table y (- x 1) n)
                            (walk-up! (+ n 1) y (- x 1) table)]
                           [(and (< (+ y 1) xynums)
                                 (not (array-ref table (+ y 1) x)))
                            (array-set! table (+ y 1) x n)
                            (walk-left! (+ n 1) (+ y 1) x table)]
                           [else table]))]
           [walk-up! (^[n y x table]
                       (cond
                         [(and (<= 0 (- y 1))
                               (not (array-ref table (- y 1) x)))
                          (array-set! table (- y 1) x n)
                          (walk-right! (+ n 1) (- y 1) x table)]
                         [(and (<= 0 (- x 1))
                               (not (array-ref table y (- x 1))))
                          (array-set! table y (- x 1) n)
                          (walk-up! (+ n 1) y (- x 1) table)]
                         [else table]))])
    (let ([table (make-array (shape 0 xynums 0 xynums) #f)])
      (array-set! table (div xynums 2) (div xynums 2) 1)
      (walk-right! 2 (div xynums 2) (div xynums 2) table))))

(define answer-28
  (apply + (diagonal-nums 1001)))

(format #t "28: ~d~%" answer-28)