model 'rosebush' ! Rosebush.mos : Computes move to safe position (if available) in game of Rosebushes ! Written by : Martin J. Chlond ! Date written : 6 January 2002 ! References : Vajda, S., Mathematical Games and how to play them (pp 59,59) ! Schuh, F., The Master Book of Mathematical Recreations (*109 pp 141,142) ! Berlekamp, E.R., Conway, J.H., Guy, R.K., Winning Ways for your Mathematical Plays uses 'mmxprs' parameters heap = 5 ! number of heaps nmax = 16 ! maximum number allowed in any heap k = 2 ! maximum number of heaps to change end-parameters declarations ! data tables n: array(1..heap) of real z: array(1..4) of real ! variables d: array(1..heap) of mpvar ! 1 if heap changed, 0 otherwise m: array(1..heap) of mpvar ! number in each heap after move e: array(1..heap) of mpvar ! 1 if heap odd after move, 0 if even b: array(1..heap) of mpvar ! dummy variable for parity check of heaps w: array(1..4,1..3) of mpvar ! dummy variables for safety check s: mpvar ! decimal equivalent of heap parities end-declarations n:=[1,1,2,3,4] ! number in each heap before move (example from Vajda) z:=[0,7,19,28] ! decimal equivalents of 'safe' heap parities (see Vajda) ! minimise number of heaps changed - if solution is zero then current position already safe heapch:= sum(i in 1..heap) d(i) ! maximum of k rows to change maxh:= sum(i in 1..heap) d(i) <= k ! ensures safe position after move forall(i in 1..4) do lca(i):= s-(32-z(i))*w(i,1) <= z(i)-1 lcb(i):= s-z(i)*w(i,1) >= 0 lcc(i):= s+(z(i)+1)*w(i,2) >= z(i)+1 lcd(i):= s+31*w(i,2) <= z(i)+31 lce(i):= w(i,3) = w(i,1)+w(i,2)-1 end-do es:= sum(i in 1..4) w(i,3) = 1 ! positions before and after are consistent with move forall(i in 1..heap) cons(i):= n(i)-d(i) = m(i) ! e(i) = 1 if m(i) odd, 0 otherwise forall(i in 1..heap) eset(i):= m(i) = 2*b(i)+e(i) ! computes decimal equivalent of heap parities sset:= sum(i in 1..heap) 2^(heap-i)*e(i) = s ! ensures piles remain in increasing order forall(i in 1..heap-1) ndec(i):= m(i+1) >= m(i) forall(i in 1..heap) do d(i) is_binary m(i) is_integer e(i) is_binary b(i) is_integer end-do forall(i in 1..4,j in 1..3) w(i,j) is_binary ! minimise number of heaps changed - if solution is zero then current position already safe minimise( heapch ) ! output solution forall(i in 1..heap) do write(getsol(d(i)),' ',getsol(m(i))) writeln end-do end-model