Railway circuit
Railway circuit
Given n sections of curve tracks, each one being an arc of 30° of radius R, the goal is to build and count all possible different railway circuits.
Constraints :
- n = 12 + k*4 (k = 0, 1 , ...)
- The circuit must be a closed, connected graph, and the last arc must joint the first one
- Duplicates, either by symmetry, translation, reflexion or rotation must be eliminated.
- Paths may overlap or cross each other.
- All tracks must be used.
Illustrations : http://www.echolalie.org/echolisp/duplo.html
Task:
Write a function which counts and displays all possible circuits Cn for n = 12, 16 , 20. Extra credit for n = 24, 28, ... 48 (no display, only counts). A circuit Cn will be displayed as a list, or sequence of n Right=1/Left=-1 turns.
Example:
C12 = (1,1,1,1,1,1,1,1,1,1,1,1) or C12 = (-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1)
Straight tracks (extra-extra credit)
Suppose we have m = k*2 sections of straight tracks, each of length L. Such a circuit is denoted Cn,m . A circuit is a sequence of +1,-1, or 0 = straight move. Count the number of circuits Cn,m with n same as above and m = 2 to 8 .
EchoLisp
<lang scheme>
- R is turn counter in right direction
- The nb of right turns in direction i
- must be = to nb of right turns in direction i+6 (opposite)
(define (legal? R) (for ((i 6)) #:break (!= (vector-ref R i) (vector-ref R (+ i 6))) => #f #t))
- equal circuits by rotation ?
(define (circuit-eq? Ca Cb) (for [(i (vector-length Cb))] #:break (eqv? Ca (vector-rotate! Cb 1)) => #t #f))
- check a result vector RV of circuits
- Remove equivalent circuits
(define (check-circuits RV) (define n (vector-length RV)) (for ((i (1- n))) #:continue (null? (vector-ref RV i)) (for ((j (in-range (1+ i) n ))) #:continue (null? (vector-ref RV j)) (when (circuit-eq? (vector-ref RV i) (vector-ref RV j)) (vector-set! RV j null)))))
- global
- *circuits* = result set = a vector
(define-values (*count* *calls* *circuits*) (values 0 0 null))
- generation of circuit C[i] i = 0 .... maxn including straight (may be 0) tracks
(define (circuits C Rct R D n maxn straight ) (define _Rct Rct) ;; save area (define _Rn (vector-ref R Rct)) (++ *calls* )
(cond
[(> *calls* 4_000_000) #f] ;; enough for maxn=24 ;; hit !! legal solution [(and (= n maxn) ( zero? Rct ) (legal? R) (legal? D))
(++ *count*) (vector-push *circuits* (vector-dup C))];; save solution
;; stop [( = n maxn) #f]
;; important cutter - not enough right turns [(and (!zero? Rct) (< (+ Rct maxn ) (+ n straight 11))) #f]
[else
;; play right (vector+= R Rct 1) ; R[Rct] += 1 (set! Rct (modulo (1+ Rct) 12)) (vector-set! C n 1) (circuits C Rct R D (1+ n) maxn straight)
;; unplay it - restore values (set! Rct _Rct) (vector-set! R Rct _Rn) (vector-set! C n '-)
;; play left (set! Rct (modulo (1- Rct) 12)) (vector-set! C n -1) (circuits C Rct R D (1+ n) maxn straight)
;; unplay (set! Rct _Rct) (vector-set! R Rct _Rn) (vector-set! C n '-)
;; play straight line (when (!zero? straight) (vector-set! C n 0) (vector+= D Rct 1) (circuits C Rct R D (1+ n) maxn (1- straight))
;; unplay (vector+= D Rct -1) (vector-set! C n '-)) ]))
- (generate max-tracks [ + max-straight])
(define (gen (maxn 20) (straight 0)) (define R (make-vector 12)) (define D (make-vector 12)) (define C (make-vector maxn '-)) (set!-values (*count* *calls* *circuits*) (values 0 0 (make-vector 0))) (vector-set! R 0 1) ;; play starter (always right) (vector-set! C 0 1) (circuits C 1 R D 1 (+ maxn straight) straight) (writeln 'gen-counters (cons *calls* *count*))
(check-circuits *circuits*) (set! *circuits* (for/vector ((c *circuits*)) #:continue (null? c) c)) (if (zero? straight) (printf "Number of circuits C%d : %d" maxn (vector-length *circuits*)) (printf "Number of circuits C%d,%d : %d" maxn straight (vector-length *circuits*))) (when (< (vector-length *circuits*) 20) (for-each writeln *circuits*))) </lang>
- Output:
(gen 12) gen-counters (331 . 1) Number of circuits C12 : 1 #( 1 1 1 1 1 1 1 1 1 1 1 1) (gen 16) gen-counters (8175 . 6) Number of circuits C16 : 1 #( 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1) (gen 20) gen-counters (150311 . 39) Number of circuits C20 : 6 #( 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 1 -1 1) #( 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 1 1) #( 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 1 -1 1) #( 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 1 1 -1 1) #( 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 -1 1) #( 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1) (gen 24) gen-counters (2574175 . 286) Number of circuits C24 : 35 (gen 12 4) Number of circuits C12,4 : 4 #( 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0) #( 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0) #( 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0) #( 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0)
zkl
<lang zkl> // R is turn counter in right direction
// The nb of right turns in direction i // must be = to nb of right turns in direction i+6 (opposite)
fcn legal(R){
foreach i in (6){ if(R[i]!=R[i+6]) return(False) } True
}
// equal circuits by rotation ?
fcn circuit_eq(Ca,Cb){
foreach i in (Cb.len()){ if(Ca==Cb.append(Cb.pop(0))) return(True) } False
}
// check a result vector RV of circuits // Remove equivalent circuits
fcn check_circuits(RV){ // modifies RV
n:=RV.len(); foreach i in (n - 1){ if(not RV[i]) continue; foreach j in ([i+1..n-1]){ if(not RV[j]) continue; if(circuit_eq(RV[i],RV[j])) RV[j]=Void; } } RV
}
// global variables // *circuits* = result set = a vector
var _count, _calls, _circuits;
// generation of circuit C[i] i = 0 .... maxn including straight (may be 0) tracks
fcn circuits([List]C,[Int]Rct,[List]R,[List]D,n,maxn, straight){
_Rct,_Rn:=Rct,R[Rct]; // save area _calls+=1;
if(_calls>0d4_000_000) False; // enough for maxn=24 else if(n==maxn and 0==Rct and legal(R) and legal(D)){ // hit legal solution _count+=1; _circuits.append(C.copy()); // save solution }else if(n==maxn) False; // stop
// important cutter - not enough right turns
else if(Rct and ((Rct + maxn) < (n + straight + 11))) False else{ // play right R[Rct]+=1; Rct=(Rct+1)%12; C[n]=1; circuits(C,Rct,R,D,n+1, maxn, straight);
Rct=_Rct; R[Rct]=_Rn; C[n]=Void; // unplay it - restore values // play left Rct=(Rct - 1 + 12)%12; C[n]=-1; // -1%12 --> 11 in EchoLisp circuits(C,Rct,R,D,n+1,maxn,straight); Rct=_Rct; R[Rct]=_Rn; C[n]=Void; // unplay if(straight){ // play straight line
C[n]=0; D[Rct]+=1; circuits(C,Rct,R,D,n+1,maxn,straight-1); D[Rct]+=-1; C[n]=Void; // unplay
} }
}
// (generate max-tracks [ + max-straight])
fcn gen(maxn=20,straight=0){
R,D:=(12).pump(List(),0), R.copy(); // vectors of zero C:=(maxn + straight).pump(List(),Void.noop); // vector of Void _count,_calls,_circuits = 0,0,List(); R[0]=C[0]=1; // play starter (always right) circuits(C,1,R,D,1,maxn + straight,straight); println("gen-counters %,d . %d".fmt(_calls,_count));
_circuits=check_circuits(_circuits).filter(); if(0==straight) println("Number of circuits C%,d : %d".fmt(maxn,_circuits.len())); else println("Number of circuits C%,d,%d : %d".fmt(maxn,straight,_circuits.len())); if(_circuits.len()<20) _circuits.apply2(T(T("toString",*),"println"));
}</lang> <lang zkl>gen(12); println(); gen(16); println(); gen(20); println(); gen(24); println(); gen(12,4);</lang>
- Output:
gen-counters 331 . 1 Number of circuits C12 : 1 L(1,1,1,1,1,1,1,1,1,1,1,1) gen-counters 8,175 . 6 Number of circuits C16 : 1 L(1,1,1,1,1,1,-1,1,1,1,1,1,1,1,-1,1) gen-counters 150,311 . 39 Number of circuits C20 : 6 L(1,1,1,1,1,1,-1,1,-1,1,1,1,1,1,1,1,-1,1,-1,1) L(1,1,1,1,1,1,-1,-1,1,1,1,1,1,1,1,1,-1,-1,1,1) L(1,1,1,1,1,1,-1,-1,1,1,1,1,1,1,1,-1,1,1,-1,1) L(1,1,1,1,1,-1,1,1,-1,1,1,1,1,1,1,-1,1,1,-1,1) L(1,1,1,1,-1,1,1,1,-1,1,1,1,1,1,-1,1,1,1,-1,1) L(1,1,1,-1,1,1,1,1,-1,1,1,1,1,-1,1,1,1,1,-1,1) gen-counters 2,574,175 . 286 Number of circuits C24 : 35 gen-counters 375,211 . 21 Number of circuits C12,4 : 4 L(1,1,1,1,1,1,0,0,1,1,1,1,1,1,0,0) L(1,1,1,1,1,0,1,0,1,1,1,1,1,0,1,0) L(1,1,1,1,0,1,1,0,1,1,1,1,0,1,1,0) L(1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0)