Re: Simple n-tuple problem - with no simple solution

*To*: mathgroup at smc.vnet.net*Subject*: [mg115879] Re: Simple n-tuple problem - with no simple solution*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Sun, 23 Jan 2011 17:32:58 -0500 (EST)

I may as well show how to convert my results into the more voluminous ones returned by other solvers. A similar conversion works if IntegerPartitions (which seems faster) is substituted for my ILP method using Solve. So here goes: Clear[t1] t1[range_List, n_Integer] := Module[{addends = Rationalize@range, multipliers, m, sum, nonNegative}, multipliers = Array[m, Length@range]; sum = Total@multipliers; nonNegative = And @@ Thread[multipliers >= 0]; Sort[multipliers /. Solve[{multipliers.addends == 1, nonNegative, sum == n}, multipliers, Integers]]] Example: n = 5; addends = Rationalize@Range[0, 1.0, 0.05]; s1 = t1[addends, n]; Length@s1 192 After removing rearrangements, ALL the solvers give 192 solutions for n = 5. Here's a randomly chosen solution. r1 = RandomChoice@s1 r2 = MapIndexed[{#1, addends[[First@#2]]} &, r1] /. {0, _} :> Sequence[] {0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} {{1, 1/20}, {2, 3/20}, {1, 3/10}, {1, 7/20}} That represents all solutions that involve 1/20 once, 3/20 twice, 3/10 once, and 7/20 once -- in any order -- for a total of n = 5 choices from the range. Now to expand the chosen solution into all its permutations: r3 = Permutations@ Flatten@Replace[r2, {k_Integer, x_} :> Table[x, {k}], 1] {{1/20, 3/20, 3/20, 3/10, 7/20}, {1/20, 3/20, 3/20, 7/20, 3/10}, {1/ 20, 3/20, 3/10, 3/20, 7/20}, {1/20, 3/20, 3/10, 7/20, 3/20}, {1/20, 3/20, 7/20, 3/20, 3/10}, {1/20, 3/20, 7/20, 3/10, 3/20}, {1/20, 3/ 10, 3/20, 3/20, 7/20}, {1/20, 3/10, 3/20, 7/20, 3/20}, {1/20, 3/10, 7/20, 3/20, 3/20}, {1/20, 7/20, 3/20, 3/20, 3/10}, {1/20, 7/20, 3/ 20, 3/10, 3/20}, {1/20, 7/20, 3/10, 3/20, 3/20}, {3/20, 1/20, 3/20, 3/10, 7/20}, {3/20, 1/20, 3/20, 7/20, 3/10}, {3/20, 1/20, 3/10, 3/ 20, 7/20}, {3/20, 1/20, 3/10, 7/20, 3/20}, {3/20, 1/20, 7/20, 3/20, 3/10}, {3/20, 1/20, 7/20, 3/10, 3/20}, {3/20, 3/20, 1/20, 3/10, 7/ 20}, {3/20, 3/20, 1/20, 7/20, 3/10}, {3/20, 3/20, 3/10, 1/20, 7/ 20}, {3/20, 3/20, 3/10, 7/20, 1/20}, {3/20, 3/20, 7/20, 1/20, 3/ 10}, {3/20, 3/20, 7/20, 3/10, 1/20}, {3/20, 3/10, 1/20, 3/20, 7/ 20}, {3/20, 3/10, 1/20, 7/20, 3/20}, {3/20, 3/10, 3/20, 1/20, 7/ 20}, {3/20, 3/10, 3/20, 7/20, 1/20}, {3/20, 3/10, 7/20, 1/20, 3/ 20}, {3/20, 3/10, 7/20, 3/20, 1/20}, {3/20, 7/20, 1/20, 3/20, 3/ 10}, {3/20, 7/20, 1/20, 3/10, 3/20}, {3/20, 7/20, 3/20, 1/20, 3/ 10}, {3/20, 7/20, 3/20, 3/10, 1/20}, {3/20, 7/20, 3/10, 1/20, 3/ 20}, {3/20, 7/20, 3/10, 3/20, 1/20}, {3/10, 1/20, 3/20, 3/20, 7/ 20}, {3/10, 1/20, 3/20, 7/20, 3/20}, {3/10, 1/20, 7/20, 3/20, 3/ 20}, {3/10, 3/20, 1/20, 3/20, 7/20}, {3/10, 3/20, 1/20, 7/20, 3/ 20}, {3/10, 3/20, 3/20, 1/20, 7/20}, {3/10, 3/20, 3/20, 7/20, 1/ 20}, {3/10, 3/20, 7/20, 1/20, 3/20}, {3/10, 3/20, 7/20, 3/20, 1/ 20}, {3/10, 7/20, 1/20, 3/20, 3/20}, {3/10, 7/20, 3/20, 1/20, 3/ 20}, {3/10, 7/20, 3/20, 3/20, 1/20}, {7/20, 1/20, 3/20, 3/20, 3/ 10}, {7/20, 1/20, 3/20, 3/10, 3/20}, {7/20, 1/20, 3/10, 3/20, 3/ 20}, {7/20, 3/20, 1/20, 3/20, 3/10}, {7/20, 3/20, 1/20, 3/10, 3/ 20}, {7/20, 3/20, 3/20, 1/20, 3/10}, {7/20, 3/20, 3/20, 3/10, 1/ 20}, {7/20, 3/20, 3/10, 1/20, 3/20}, {7/20, 3/20, 3/10, 3/20, 1/ 20}, {7/20, 3/10, 1/20, 3/20, 3/20}, {7/20, 3/10, 3/20, 1/20, 3/ 20}, {7/20, 3/10, 3/20, 3/20, 1/20}} As you can see, sixty solutions from other solvers are represented by ONE solution from "t1" (for this particular randomly chosen solution). Length@r3 60 Here's the code that removes rearrangements, which I used to verify that other solvers were returning the right number of DISTINCT solutions. Union[Sort /@ r3] {{1/20, 3/20, 3/20, 3/10, 7/20}} Bobby -- DrMajorBob at yahoo.com