perplexus.info :: Algorithms : Intuitive Coins

Intuitive Coins (Posted on 2005-03-29)

If you must pay an amount in coins, the "intuitive" algorithm is: pay as much as possible with the largest denomination coin, and then go on to pay the rest with the other coins. For example, if there are 25, 5 and 1 cent coins, to pay someone 32 cents, you'd first give him a 25 cents coin, then one 5 cent coin, and finally two 1 cent coins.)

However, this doesn't always end paying with as few coins as possible: if we had 25, 10 and 1 cent coins, paying 32 cents with the "intuitive" algorithm would use 8 coins, while three 10 cent coins and two 1 cent coins would be better.

We can call a set "intuitive", if the "intuitive algorithm" always pays out any amount with as few coins as possible.

The problem: give an algorithm that allows you to decide that {25,5,1} is an "intuitive" set, while {25,10,1} isn't.

Submitted by Federico Kereki

Rating: 3.8000 (5 votes)

Solution: (Hide)

For sets with three coins ({1,c,d} with c<d) Kozen and Zaks prove that the set isn't intuitive (canonical) if and only if d=qc+r (0≤c<r) and 0<r<c-q. This test proves {1,5,25} is intuitive, but {1,10,25} isn't.
For the general problem, there's no easy answer. A sufficient (but not necessary condition) is that if each coin is a multiple of the next smallest, the set is intuitive; a special case is any set {1, p, p², p³...}.
A group of interesting references is http://mathforum.org/kb/message.jspa?messageID=138837&tstart=0 and solutions can be found at www.cs.cornell.edu/kozen/papers/change.ps and http://citeseer.ist.psu.edu/pearson94polynomialtime.html

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
inefficient but foolproof	steve	2005-04-08 04:07:31
re: On the right track	armando	2005-04-02 22:25:23
On the right track	Steve Herman	2005-04-02 14:49:14
idea	armando	2005-04-02 09:32:01
re(3): There's more to it.... (solution? spoiler?) -- larger sets	Charlie	2005-03-31 14:18:05
re(2): There's more to it.... (solution? spoiler?) -- larger sets	Charlie	2005-03-31 13:08:57
re: There's more to it.... (solution? spoiler?) -- larger sets	Charlie	2005-03-30 19:40:13
There's more to it.... (solution? spoiler?)	Erik O.	2005-03-30 15:47:52
intuitive enough	armando	2005-03-30 13:21:39
re: quick thought	Avin	2005-03-30 13:19:37
quick thought	Larry	2005-03-30 12:49:24
re(2): Ideas	Steve Herman	2005-03-29 22:39:55
re: Ideas	John	2005-03-29 20:58:14
Ideas	Gamer	2005-03-29 19:39:11

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