Imagine seeking a path from a given source to given destination in a Manhattan-like city grid. How many path there are to arrive to point a to point b?
This variant assumes that that the grid is a 3D space
ANSWER:
(define manhattan-3d
(lambda (i j k)
(cond
((and (= j 0) (= k 0)) 1)
((and (= j 0) (= i 0)) 1)
((and (= k 0) (= i 0)) 1)
((= i 0) (+ (manhattan-3d i (- j 1) k)
(manhattan-3d i j (- k 1))))
((= k 0) (+ (manhattan-3d i (- j 1) k)
(manhattan-3d (- i 1) j k)))
((= j 0) (+ (manhattan-3d i j (- k 1))
(manhattan-3d (- i 1) j k)))
(else
(+ (manhattan-3d i (- j 1) k)
(manhattan-3d (- i 1) j k)
(manhattan-3d i j (- k 1))))
)
)
)
(manhattan-3d 0 0 7) ;1 (manhattan-3d 2 0 2) ;6 (manhattan-3d 1 1 1) ;6 (manhattan-3d 1 1 5) ;42 (manhattan-3d 2 3 1) ;60 (manhattan-3d 2 3 3) ;560