Exercise 2.19
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)
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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-firstdenomination, 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?
這道題題目很長，看著挺唬人的，但實際上非常簡單，只要把 coin-values 當作列表來處理就行了。
(define (no-more? items)
(null? items))
(define (except-first-denomination items)
(cdr items))
(define (first-denomination items)
(car items))
(define us-coins (list 50 25 10 5 1))
(define uk-coins (list 100 50 20 10 5 2 1 0.5))
(cc 100 us-coins)
(cc 100 uk-coins)
; 借用上道題的列表翻轉函式
(cc 100 (reverse us-coins))
(cc 100 (reverse uk-coins))
; 執行結果
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104561
292
104561
從執行結果來看，零錢幣值的順序是不影響兌換的方法數的，這是因為 cc 演算法覆蓋了所有可能的情況，無論從哪種幣值先開始兌換，都不會影響總的結果。