SICP Exercise 2.19: Counting UK Change

Consider the change-counting program of section 1.2.2. It would be nice to be able to easily change the currency used by the program, so that we could compute the number of ways to change a British pound, for example. As the program is written, the knowledge of the currency is distributed partly into the procedure first-denomination and partly into the procedure count-change (which knows that there are five kinds of U.S. coins). It would be nicer to be able to supply a list of coins to be used for making change.

We want to rewrite the procedure cc so that its second argument is a list of the values of the coins to use rather than an integer specifying which coins to use. We could then have lists that defined each kind of currency:

(define us-coins (list 50 25 10 5 1))
(define uk-coins (list 100 50 20 10 5 2 1 0.5))

We could then call cc as follows:

> (cc 100 us-coins)
292

To do this will require changing the program cc somewhat. It will still have the same form, but it will access its second argument differently, as follows:

(define (cc amount coin-values)
  (cond ((= amount 0) 1)
        ((or (< amount 0) (no-more? coin-values)) 0)
        (else
         (+ (cc amount
                (except-first-denomination coin-values))
            (cc (- amount
                   (first-denomination coin-values))
                coin-values)))))

Define the procedures first-denomination, except-first-denomination, and no-more? in terms of primitive operations on list structures. Does the order of the list coin-values affect the answer produced by cc? Why or why not?

The exercise already tells us the representation we're going to be using for denominations: lists of the denomination values. As a result, the procedures first-denomination, except-first-denomination, and no-more? are straightforward to define. first-denomination just needs to return the car of the list, except-first-denomination just needs to return the cdr of the list, and no-more? just needs to check to see if the list is empty:

(define (first-denomination coin-values)
  (car coin-values))

(define (except-first-denomination coin-values)
  (cdr coin-values))

(define (no-more? coin-values)
  (null? coin-values))

We can check this works as expected:

> (cc 100 us-coins)
292
> (cc 100 uk-coins)
104561

As for whether or not the order of the list coin-values affects the answer produced by cc, we can easily test this:

> (cc 100 (list 1 5 10 25 50))
292
> (cc 100 (list 5 25 50 10 1))
292

So no, it doesn't appear as if it does affect ordering. So why not? Well, looking at the algorithm we can see that each non-terminal call to cc sums the results of two (also potentially non-terminal) recursive calls:

• One of these finds the number of ways we can change amount using denominations other than the first one in the list of denominations. As a result, this counts all of the combinations for changing amount that do not use the first denomination.
• The other deducts the value of the first denomination in the list of denominations from amount and finds how many ways we can change that using any and all of the denominations in the list. As a result, this counts all of the combinations for changing amount that do use the first denomination.

In other words, these two recursive calls split the counting to cover two distinct sets of coin combinations which, when merged together, produces a set containing all valid combinations of denominations for changing amount. As a result the order of the denominations does not matter.