10230

# Optimality of Greedy Change-Making

The U.S. coin set of 1, 5, 10, 25, 50, 100 satisfies the greedy condition, meaning that if you make change for an amount greedily (always choosing the largest coin that fits in the amount left) you get a representation of A that uses the fewest possible coins. The old British system based on the halfpenny as the unit corresponds to coins 1, 2, 6, 12, 24, 48, 60, and that system is not greedy: 96 = 48 + 48 but the greedy method gets 96 = 60 + 24 + 12. An efficient algorithm due to Pearson determines whether a set satisfies the greedy condition or not, and if not, finds the smallest target amount that demonstrates the failure. The sliders in the Demonstration control the coin differences, so to see the U.S. system, set to 6 and the differences to 4, 5, 15, 25, 50.

### DETAILS

For more details on Pearson's algorithm see
D. Pearson, "A Polynomial-Time Algorithm for the Change-Making Problem," Operations Research Letters, 33(3), 2005 pp. 231–234.

### PERMANENT CITATION

Contributed by: Stan Wagon (Macalester College)
 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.