examples/notebook/contrib/sicherman_dice.ipynb
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First, you must install ortools package in this colab.
%pip install ortools
Sicherman Dice in Google CP Solver.
From http://en.wikipedia.org/wiki/Sicherman_dice "" Sicherman dice are the only pair of 6-sided dice which are not normal dice, bear only positive integers, and have the same probability distribution for the sum as normal dice.
The faces on the dice are numbered 1, 2, 2, 3, 3, 4 and 1, 3, 4, 5, 6, 8. ""
I read about this problem in a book/column by Martin Gardner long time ago, and got inspired to model it now by the WolframBlog post "Sicherman Dice": http://blog.wolfram.com/2010/07/13/sicherman-dice/
This model gets the two different ways, first the standard way and then the Sicherman dice:
x1 = [1, 2, 2, 3, 3, 4] x2 = [1, 3, 4, 5, 6, 8]
Extra: If we also allow 0 (zero) as a valid value then the following two solutions are also valid:
x1 = [0, 1, 2, 3, 4, 5] x2 = [2, 3, 4, 5, 6, 7]
These two extra cases are mentioned here: http://mathworld.wolfram.com/SichermanDice.html
Compare with these models:
This model was created by Hakan Kjellerstrand ([email protected]) Also see my other Google CP Solver models: http://www.hakank.org/google_or_tools/
import sys
from ortools.constraint_solver import pywrapcp
def main():
# Create the solver.
solver = pywrapcp.Solver("Sicherman dice")
#
# data
#
n = 6
m = 10
# standard distribution
standard_dist = [1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]
#
# declare variables
#
# the two dice
x1 = [solver.IntVar(0, m, "x1(%i)" % i) for i in range(n)]
x2 = [solver.IntVar(0, m, "x2(%i)" % i) for i in range(n)]
#
# constraints
#
# [solver.Add(standard_dist[k] == solver.Sum([x1[i] + x2[j] == k+2 for i in range(n) for j in range(n)]))
# for k in range(len(standard_dist))]
for k in range(len(standard_dist)):
tmp = [solver.BoolVar() for i in range(n) for j in range(n)]
for i in range(n):
for j in range(n):
solver.Add(tmp[i * n + j] == solver.IsEqualCstVar(x1[i] + x2[j], k + 2))
solver.Add(standard_dist[k] == solver.Sum(tmp))
# symmetry breaking
[solver.Add(x1[i] <= x1[i + 1]) for i in range(n - 1)],
[solver.Add(x2[i] <= x2[i + 1]) for i in range(n - 1)],
[solver.Add(x1[i] <= x2[i]) for i in range(n - 1)],
#
# solution and search
#
solution = solver.Assignment()
solution.Add(x1)
solution.Add(x2)
# db: DecisionBuilder
db = solver.Phase(x1 + x2, solver.INT_VAR_SIMPLE, solver.ASSIGN_MIN_VALUE)
solver.NewSearch(db)
num_solutions = 0
while solver.NextSolution():
print("x1:", [x1[i].Value() for i in range(n)])
print("x2:", [x2[i].Value() for i in range(n)])
print()
num_solutions += 1
solver.EndSearch()
print()
print("num_solutions:", num_solutions, "solver.solutions:",
solver.Solutions())
print("failures:", solver.Failures())
print("branches:", solver.Branches())
print("WallTime:", solver.WallTime())
print("MemoryUsage:", solver.MemoryUsage())
print("SearchDepth:", solver.SearchDepth())
print("SolveDepth:", solver.SolveDepth())
print("stamp:", solver.Stamp())
print("solver", solver)
main()