**change_polynomial**,
a Python code which
uses a polynomial multiplication algorithm to count the ways of making
various sums using a given number of coins.

In this version of the change making problem, the order of the coins is significant. In other words, if we are using exactly 3 coins to make 16 cents, then we count 6 separate solutions: (1,5,10), (1,10,5), (5,1,10), (5,10,1), (10,1,5) and (10,5,1). Similarly, we count three essentialy equivalent solutions to forming the sum of 7 cents.

If the user wishes to determine the number of ways of forming sums using anywhere from 0 up to COIN_NUM coins, inclusive, then it is only necessary to include a coin of value 0 in the denomination list.

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

**change_polynomial** is available in
a MATLAB version and
an Octave version and
a Python version.

change_diophantine, a Python code which sets up a Diophantine equation to solve the change making problem, which counts the number of ways a given sum can be formed using coins of various denominations.

change_dynamic, a Python code which uses dynamic programming to solve the change making problem, which counts the number of ways a given sum can be formed using coins of various denominations.

change_greedy, a Python code which uses the greedy method to seek a solution to the change making problem, which tries to match a given amount by selecting coins of various denominations.

combination_lock, a Python code which simulates the process of determining the secret combination of a lock.

combo, a Python code which includes many combinatorial routines.

football_dynamic, a Python code which uses dynamic programming to count the ways of achieving a given score in football.

knapsack_01, a Python code which uses brute force to solve small versions of the 0/1 knapsack problem;

mcnuggets, a Python code which counts M(N), the number of ways a given number N of Chicken McNuggets can be assembled, given that they are only available in packages of 6, 9, and 20.

partition_brute, a Python code which uses brute force to seek solutions of the partition problem, splitting a set of integers into two subsets with equal sum.

partition_greedy, a Python code which uses a greedy algorithm to seek a solution of the partition problem, in which a given set of integers is to be split into two groups whose sums are as close as possible.

polynomial_multiply, a Python code which multiplies two polynomials p(x) and q(x).

satisfy, a Python code which demonstrates, for a particular circuit, an exhaustive search for solutions of the circuit satisfiability problem.

subset, a Python code which enumerates, generates, ranks and unranks combinatorial objects including combinations, partitions, subsets, index sets, and trees.

subset_sum, a Python code which seeks solutions of the subset sum problem, in which it is desired to find a subset of a set of integers which has a given sum.

subset_sum_brute, a Python code which uses brute force to solve the subset sum problem, to find a subset of a set of integers which has a given sum.

tsp_brute, a Python code which is given a city-to-city distance table, and solves a (small) traveling salesperson problem (TSP), using brute force.

tsp_greedy, a Python code which is given a city-to-city distance table,, and solves a small traveling salesperson problem (TSP) using the greedy algorithm. It picks a starting city at random, and then successively visits the nearest unvisited city.

- change_polynomial.py, the source code.
- change_polynomial.sh, runs all the tests.
- change_polynomial.txt, the output file.