# Code written by Andy Huchala # Computes a(n) for OEIS A279402 # (the minimum queens required to occupy or # threaten all tiles on a toroidal n x n chessboard) # Requires installing Gurobi # Select board size (n>1) n = 8 from gurobipy import * m = Model("ip") # initialize all variables of form x_j_i for i in range(n): for j in range(n): exec("x_" + str(j) + "_" + str(i)+" = m.addVar(lb=0,ub=1,vtype=GRB.INTEGER, name=\"x_" + str(j) + "_" + str(i) + "\")") # Set objective: minimize sum of x_i_j's t = "x_0_0" for j in range(n): for i in range(n): if i + j != 0: t += "+x_" + str(j) + "_" + str(i) exec("obj = " + t) m.setObjective(obj, GRB.MINIMIZE) # specify constraints for j in range(n): for i in range(n): # find all the locations from which (i,j) could be attacked, add each one to the constraint # for (i,j): (i,j) must be attacked or occupied s = "m.addLConstr(" s += "x_" + str(j) + "_" + str(i) + "+" for k in range(n): if k != j: s += "x_" + str(k) + "_" + str(i) + "+" if k != i: s += "x_" + str(j) + "_" + str(k) + "+" if k != j or (i-j+k)%n != i: s += "x_" + str(k) + "_" + str((i-j+k)%n) + "+" if k != j or (2*i-(i-j+k))%n != i: s += "x_" + str(k) + "_" + str((2*i-(i-j+k))%n) + "+" s = s[:-1] exec(s+ ">=1, \"" + "c_" + str(j) + "_" + str(i) + "\")") m.optimize() # for v in m.getVars(): # print('%s %g' % (v.varName, v.x)) print('Obj: %g' % obj.getValue())