sicp: section 1.2

sicp/1/2/1.scm

+(define (factorial n)
+  (if (= n 1)
+    1
+    (* n (factorial (- n 1)))))
+
+(display (factorial 6))
+(newline)
+
+(define (factorial n)
+  (fact-iter 1 1 n))
+
+(define (fact-iter product counter max-count)
+  (if (> counter max-count)
+    product
+    (fact-iter (* counter product)
+               (+ counter 1)
+               max-count)))
+
+(display (factorial 6))
+(newline)

sicp/1/2/2.scm

+(define (fib n)
+  (cond ((= n 0) 0)
+        ((= n 1) 1)
+        (else (+ (fib (- n 1))
+                 (fib (- n 2))))))
+(display (fib 5))
+(newline)
+
+(define (fib n)
+  (fib-iter 1 0 n))
+(define (fib-iter a b count)
+  (if (= count 0)
+    b
+    (fib-iter (+ a b) a (- count 1))))
+(display (fib 5))
+(newline)
+
+(define (count-change amount)
+  (cc amount 5))
+(define (cc amount kinds-of-coins)
+  (cond ((= amount 0) 1)
+        ((or (< amount 0) (= kinds-of-coins 0)) 0)
+        (else (+ (cc amount
+                     (- kinds-of-coins 1))
+                 (cc (- amount
+                        (first-denomination kinds-of-coins))
+                     kinds-of-coins)))))
+(define (first-denomination kinds-of-coins)
+  (cond ((= kinds-of-coins 1) 1)
+        ((= kinds-of-coins 2) 5)
+        ((= kinds-of-coins 3) 10)
+        ((= kinds-of-coins 4) 25)
+        ((= kinds-of-coins 5) 50)))
+(display (count-change 100))

sicp/1/2/4.scm

+(define (test)
+  (display (expt 2 1))
+  (display " ")
+  (display (expt 2 2))
+  (display " ")
+  (display (expt 2 3))
+  (display " ")
+  (display (expt 2 4))
+  (display " ")
+  (display (expt 2 5))
+  (display " ")
+  (display (expt 2 6))
+  (display " ")
+  (display (expt 2 7))
+  (display " ")
+  (display (expt 2 8))
+  (display " ")
+  (display (expt 2 9))
+  (display " ")
+  (display (expt 2 10))
+  (newline))
+
+(define (expt b n)
+  (if (= n 0)
+    1
+    (* b (expt b (- n 1)))))
+
+(test)
+
+(define (expt b n)
+  (define (expt-iter b counter product)
+    (if (= counter 0)
+      product
+      (expt-iter b
+                 (- counter 1)
+                 (* b product))))
+  (expt-iter b n 1))
+
+(test)
+
+(define (expt b n)
+  (cond ((= n 0) 1)
+        ((even? n) (square (expt b (/ n 2))))
+        (else (* b (expt b (- n 1))))))
+
+(test)

sicp/1/2/5.scm

+(define (gcd a b)
+  (if (= b 0)
+    a
+    (gcd b (remainder a b))))
+
+(display (gcd 206 40))
+(newline)

sicp/1/2/6.scm

+(define (smallest-divisor n)
+  (define (find-divisor n test-divisor)
+    (define (divides? a b)
+      (= (remainder b a) 0))
+    (cond ((> (square test-divisor) n) n)
+          ((divides? test-divisor n) test-divisor)
+          (else (find-divisor n (+ test-divisor 1)))))
+  (find-divisor n 2))
+
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(define (test prime?)
+  (define (iter n)  (display " ") (display (prime? n)))
+  (display "non-primes:")
+  (map iter '(561 1105 1729 2465 2821 6601))
+  (newline)
+  (display "primes:")
+  (map iter '(2 5 13 37 1009 10009)))
+(test prime?)
+
+(define (expmod base exp m)
+  (cond ((= exp 0) 1)
+        ((even? exp)
+         (remainder (square (expmod base (/ exp 2) m))
+                    m))
+        (else
+          (remainder (* base (expmod base (- exp 1) m))
+                     m))))
+
+(define (fermat-test n)
+  (define (try-it a)
+    (= (expmod a n n) a))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (fast-prime? n times)
+  (cond ((= times 0) true)
+        ((fermat-test n) (fast-prime? n (- times 1)))
+        (else false)))
+
+(newline)
+(newline)
+(test (lambda (n) (fast-prime? n 100)))

sicp/1/2/ex_10.scm

+(define (A x y)
+  (cond ((= y 0) 0)
+        ((= x 0) (* 2 y))
+        ((= y 1) 2)
+        (else (A (- x 1)
+                 (A x (- y 1))))))
+(display (A 1 10))
+(newline)
+(display (A 2 4))
+(newline)
+(display (A 3 3))

sicp/1/2/ex_11.scm

+(define (f n)
+  (if (< n 3)
+    n
+    (+ (f (- n 1))
+       (* 2 (f (- n 2)))
+       (* 3 (f (- n 3))))))
+(display (f 5))
+(newline)
+
+(define (f n)
+  (if (< n 3)
+    n
+    (f-iter 0 1 2 (- n 3))))
+(define (f-iter f-3 f-2 f-1 count)
+  (define new-f (+ f-1 (* 2 f-2) (* 3 f-3)))
+  (if (= count 0)
+    new-f
+    (f-iter f-2 f-1 new-f (- count 1))))
+(display (f 5))
+(newline)

sicp/1/2/ex_12.scm

+(define (elem-xy x y)
+  (if (or (= x 0) (= x y))
+    1
+    (+ (elem-xy (- x 1) (- y 1))
+       (elem-xy x (- y 1)))))
+
+(display (elem-xy 0 0))
+(newline)
+(display (elem-xy 0 1))
+(display " ")
+(display (elem-xy 1 1))
+(newline)
+(display (elem-xy 0 2))
+(display " ")
+(display (elem-xy 1 2))
+(display " ")
+(display (elem-xy 2 2))
+(newline)
+(display (elem-xy 0 3))
+(display " ")
+(display (elem-xy 1 3))
+(display " ")
+(display (elem-xy 2 3))
+(display " ")
+(display (elem-xy 3 3))
+(newline)
+(display (elem-xy 0 4))
+(display " ")
+(display (elem-xy 1 4))
+(display " ")
+(display (elem-xy 2 4))
+(display " ")
+(display (elem-xy 3 4))
+(display " ")
+(display (elem-xy 4 4))
+(newline)

sicp/1/2/ex_16.scm

+(define (expt b n)
+  (define (expt-iter b n a)
+    (cond ((= n 0) a)
+          ((odd? n) (expt-iter b (- n 1) (* b a)))
+          (else (expt-iter (* b b) (/ n 2) a))))
+  (expt-iter b n 1))
+
+(display (expt 2 1))
+(display " ")
+(display (expt 2 2))
+(display " ")
+(display (expt 2 3))
+(display " ")
+(display (expt 2 4))
+(display " ")
+(display (expt 2 5))
+(display " ")
+(display (expt 2 6))
+(display " ")
+(display (expt 2 7))
+(display " ")
+(display (expt 2 8))
+(display " ")
+(display (expt 2 9))
+(display " ")
+(display (expt 2 10))
+(newline)

sicp/1/2/ex_17.scm

+(define (book-* a b)
+  (if (= b 0)
+    0
+    (+ a (* a (- b 1)))))
+
+(define (fast-mul a b)
+  (define (double x)
+    (+ x x))
+  (define (havle x)
+    (/ x 2))
+  (cond ((= b 0) 0)
+        ((even? b) (double (fast-mul a (/ b 2))))
+        (else (+ a (fast-mul a (- b 1))))))
+
+(map
+  (lambda (x)
+    (let
+      ((book2 (book-* 2 x)) (impl2 (fast-mul 2 x))
+       (book3 (book-* 3 x)) (impl3 (fast-mul 3 x))
+       (book4 (book-* 4 x)) (impl4 (fast-mul 4 x)))
+      (if (< book2 10) (display " "))
+      (display book2)
+      (display " = ")
+      (if (< impl2 10) (display " "))
+      (display impl2)
+      (display "  ")
+      (if (< book3 10) (display " "))
+      (display book3)
+      (display " = ")
+      (if (< impl3 10) (display " "))
+      (display impl3)
+      (display "  ")
+      (if (< book4 10) (display " "))
+      (display book4)
+      (display " = ")
+      (if (< impl4 10) (display " "))
+      (display impl4)
+      (newline)))
+  '(1 2 3 4 5 6 7 8 9 10))

sicp/1/2/ex_18.scm

+(define (book-* a b)
+  (if (= b 0)
+    0
+    (+ a (* a (- b 1)))))
+
+(define (fast-mul a b)
+  (define (double x)
+    (+ x x))
+  (define (havle x)
+    (/ x 2))
+  (define (*-iter b n a)
+    (cond ((= n 0) a)
+          ((odd? n) (*-iter b (- n 1) (+ b a)))
+          (else (*-iter (double b) (/ n 2) a))))
+  (*-iter b a 0))
+
+(map
+  (lambda (x)
+    (let
+      ((book2 (book-* 2 x)) (impl2 (fast-mul 2 x))
+       (book3 (book-* 3 x)) (impl3 (fast-mul 3 x))
+       (book4 (book-* 4 x)) (impl4 (fast-mul 4 x)))
+      (if (< book2 10) (display " "))
+      (display book2)
+      (display " = ")
+      (if (< impl2 10) (display " "))
+      (display impl2)
+      (display "  ")
+      (if (< book3 10) (display " "))
+      (display book3)
+      (display " = ")
+      (if (< impl3 10) (display " "))
+      (display impl3)
+      (display "  ")
+      (if (< book4 10) (display " "))
+      (display book4)
+      (display " = ")
+      (if (< impl4 10) (display " "))
+      (display impl4)
+      (newline)))
+  '(1 2 3 4 5 6 7 8 9 10))

sicp/1/2/ex_19.scm

+(define (fib n)
+  (fib-iter 1 0 0 1 n))
+(define (fib-iter a b p q count)
+  (cond ((= count 0) b)
+        ((even? count)
+         (fib-iter a
+                   b
+                   (+ (* p p) (* q q))
+                   (+ (* q q) (* 2 p q))
+                   (/ count 2)))
+        (else (fib-iter (+ (* b q) (* a q) (* a p))
+                        (+ (* b p) (* a q))
+                        p
+                        q
+                        (- count 1)))))
+(display (fib 0))
+(newline)
+(display (fib 1))
+(newline)
+(display (fib 2))
+(newline)
+(display (fib 3))
+(newline)
+(display (fib 4))
+(newline)
+(display (fib 5))
+(newline)
+(display (fib 6))
+(newline)
+(display (fib 7))
+(newline)
+(display (fib 8))

sicp/1/2/ex_21.scm

+(define (smallest-divisor n)
+  (define (find-divisor n test-divisor)
+    (define (divides? a b)
+      (= (remainder b a) 0))
+    (cond ((> (square test-divisor) n) n)
+          ((divides? test-divisor n) test-divisor)
+          (else (find-divisor n (+ test-divisor 1)))))
+  (find-divisor n 2))
+
+(display (smallest-divisor 199))
+(newline)
+(display (smallest-divisor 1999))
+(newline)
+(display (smallest-divisor 19999))
+(newline)

sicp/1/2/ex_22.scm

+(define (smallest-divisor n)
+  (define (find-divisor n test-divisor)
+    (define (divides? a b)
+      (= (remainder b a) 0))
+    (cond ((> (square test-divisor) n) n)
+          ((divides? test-divisor n) test-divisor)
+          (else (find-divisor n (+ test-divisor 1)))))
+  (find-divisor n 2))
+
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(define (timed-prime-test n)
+  (newline)
+  (display n)
+  (start-prime-test n (runtime)))
+
+(define (start-prime-test n start-time)
+  (if (prime? n)
+    (report-prime (- (runtime) start-time))))
+
+(define (report-prime elapsed-time)
+  (display " *** ")
+  (display elapsed-time))
+
+(define (search-for-primes start end)
+  (define (search-for-primes-iter i)
+    (cond ((<= i end)
+      (timed-prime-test i)
+      (search-for-primes-iter (+ i 1)))))
+  (search-for-primes-iter start))
+
+(search-for-primes 3 1000)

sicp/1/2/ex_23.scm

+(define (smallest-divisor n)
+  (define (find-divisor n test-divisor)
+    (define (divides? a b)
+      (= (remainder b a) 0))
+    (cond ((> (square test-divisor) n) n)
+          ((divides? test-divisor n) test-divisor)
+          (else (find-divisor n (next test-divisor)))))
+  (find-divisor n 2))
+
+(define (next x)
+  (if (= x 2) 3 (+ x 2)))
+
+(define (prime? n)
+  (= n (smallest-divisor n)))
+
+(define (timed-prime-test n)
+  (newline)
+  (display n)
+  (start-prime-test n (runtime)))
+
+(define (start-prime-test n start-time)
+  (if (prime? n)
+    (report-prime (- (runtime) start-time))))
+
+(define (report-prime elapsed-time)
+  (display " *** ")
+  (display elapsed-time))
+
+(define (search-for-primes start end)
+  (define (search-for-primes-iter i)
+    (cond ((<= i end)
+      (timed-prime-test i)
+      (search-for-primes-iter (+ i 1)))))
+  (search-for-primes-iter start))
+
+(search-for-primes 3 1000)

sicp/1/2/ex_24.scm

+(define (expmod base exp m)
+  (cond ((= exp 0) 1)
+        ((even? exp)
+         (remainder (square (expmod base (/ exp 2) m))
+                    m))
+        (else
+          (remainder (* base (expmod base (- exp 1) m))
+                     m))))
+
+(define (fermat-test n)
+  (define (try-it a)
+    (= (expmod a n n) a))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (fast-prime? n times)
+  (cond ((= times 0) true)
+        ((fermat-test n) (fast-prime? n (- times 1)))
+        (else false)))
+
+(define (timed-prime-test n)
+  (newline)
+  (display n)
+  (start-prime-test n (runtime)))
+
+(define (start-prime-test n start-time)
+  (if (fast-prime? n 100)
+    (report-prime (- (runtime) start-time))))
+
+(define (report-prime elapsed-time)
+  (display " *** ")
+  (display elapsed-time))
+
+(define (search-for-primes start end)
+  (define (search-for-primes-iter i)
+    (cond ((<= i end)
+      (timed-prime-test i)
+      (search-for-primes-iter (+ i 1)))))
+  (search-for-primes-iter start))
+
+(search-for-primes 3 1000)

sicp/1/2/ex_27.scm

+(define (expmod base exp m)
+  (cond ((= exp 0) 1)
+        ((even? exp)
+         (remainder (square (expmod base (/ exp 2) m))
+                    m))
+        (else
+          (remainder (* base (expmod base (- exp 1) m))
+                     m))))
+
+(define (fermat-test n)
+  (define (try-it a)
+    (= (expmod a n n) a))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (test-charmichael n)
+  (define (test-charmichael-iter i)
+    (cond ((= i n) true)
+          ((fermat-test n) (test-charmichael-iter (+ i 1)))
+          (else false)))
+  (test-charmichael-iter 0))
+
+(display (test-charmichael 561))
+(newline)
+(display (test-charmichael 1105))
+(newline)
+(display (test-charmichael 1729))
+(newline)
+(display (test-charmichael 2465))
+(newline)
+(display (test-charmichael 2821))
+(newline)
+(display (test-charmichael 6601))
+(newline)

sicp/1/2/ex_28.scm

+(define (expmod base exp m)
+  (define (squaremod-with-check x)
+    (define (check-nontrivial x s)
+      (if (and (= s 1) (not (= x 1)) (not (= x (- m 1))))
+           0 s))
+    (check-nontrivial x (remainder (square x) m)))
+  (cond ((= exp 0) 1)
+        ((even? exp) (squaremod-with-check (expmod base (/ exp 2) m)))
+        (else (remainder (* base (expmod base (- exp 1) m)) m))))
+
+(define (miller-rabin-test n)
+  (define (try-it a)
+    (define (test-expmod x)
+      (and (not (= x 0)) (= x 1)))
+    (test-expmod (expmod a (- n 1) n)))
+  (try-it (+ 1 (random (- n 1)))))
+
+(define (fast-prime? n)
+  (define (test-prime-iter i)
+    (cond ((= i 0) true)
+          ((miller-rabin-test n) (test-prime-iter (- i 1)))
+          (else false)))
+  (test-prime-iter 100))
+
+(display "non-primes: ")
+(display (fast-prime? 561))
+(display " ")
+(display (fast-prime? 1105))
+(display " ")
+(display (fast-prime? 1729))
+(display " ")
+(display (fast-prime? 2465))
+(display " ")
+(display (fast-prime? 2821))
+(display " ")
+(display (fast-prime? 6601))
+(newline)
+
+(display "primes: ")
+(display (fast-prime? 2))
+(display " ")
+(display (fast-prime? 5))
+(display " ")
+(display (fast-prime? 13))
+(display " ")
+(display (fast-prime? 37))
+(display " ")
+(display (fast-prime? 1009))
+(display " ")
+(display (fast-prime? 10009))
+(newline)

sicp/1/2/ex_9.scm

+(define (inc x) (+ x 1))
+(define (dec x) (- x 1))
+(define (add a b)
+  (if (= a 0)
+      b
+      (inc (add (dec a) b))))
+(display (add 2 7))
+(define (add a b)
+  (if (= a 0)
+      b
+      (add (dec a) (inc b))))
+(newline)
+(display (add 2 7))