Exercise 3.17. Devise a correct version of the count-pairs procedure of exercise 3.16 that returns the number of distinct pairs in any structure. (Hint: Traverse the structure, maintaining an auxiliary data structure that is used to keep track of which pairs have already been counted.)
Exercise 3.7. Consider the bank account objects created by make-account, with the password modification described in exercise 3.3. Suppose that our banking system requires the ability to make joint accounts. Define a procedure make-joint that accomplishes this. Make-joint should take three arguments. The first is a password-protected account. The second argument must match the password with which the account was defined in order for the make-joint operation to proceed. The third argument is a new password. Make-joint is to create an additional access to the original account using the new password. For example, if peter-acc is a bank account with password open-sesame, then
(define paul-acc
(make-joint peter-acc 'open-sesame 'rosebud))
will allow one to make transactions on peter-acc using the name paul-acc and the password rosebud. You may wish to modify your solution to exercise 3.3 to accommodate this new feature.
Exercise 3.6. It is useful to be able to reset a random-number generator to produce a sequence starting from a given value. Design a new rand procedure that is called with an argument that is either the symbol generate or the symbol reset and behaves as follows: (rand 'generate) produces a new random number; ((rand 'reset) <new-value>) resets the internal state variable to the designated <new-value>. Thus, by resetting the state, one can generate repeatable sequences. These are very handy to have when testing and debugging programs that use random numbers.
Exercise 3.5. Monte Carlo integration is a method of estimating definite integrals by means of Monte Carlo simulation. Consider computing the area of a region of space described by a predicate P(x, y) that is true for points (x, y) in the region and false for points not in the region. For example, the region contained within a circle of radius 3 centered at (5, 7) is described by the predicate that tests whether (x – 5)2 + (y – 7)2< 32. To estimate the area of the region described by such a predicate, begin by choosing a rectangle that contains the region. For example, a rectangle with diagonally opposite corners at (2, 4) and (8, 10) contains the circle above. The desired integral is the area of that portion of the rectangle that lies in the region. We can estimate the integral by picking, at random, points (x,y) that lie in the rectangle, and testing P(x, y) for each point to determine whether the point lies in the region. If we try this with many points, then the fraction of points that fall in the region should give an estimate of the proportion of the rectangle that lies in the region. Hence, multiplying this fraction by the area of the entire rectangle should produce an estimate of the integral.
Implement Monte Carlo integration as a procedure estimate-integral that takes as arguments a predicate P, upper and lower bounds x1, x2, y1, and y2 for the rectangle, and the number of trials to perform in order to produce the estimate. Your procedure should use the same monte-carlo procedure that was used above to estimate 
. Use your estimate-integral to produce an estimate of by measuring the area of a unit circle.
You will find it useful to have a procedure that returns a number chosen at random from a given range. The following random-in-range procedure implements this in terms of the random procedure used in section 1.2.6, which returns a nonnegative number less than its input.8
(define (random-in-range low high)
(let ((range (- high low)))
(+ low (random range))))
Exercise 3.4. Modify the make-account procedure of exercise 3.3 by adding another local state variable so that, if an account is accessed more than seven consecutive times with an incorrect password, it invokes the procedure call-the-cops.
Exercise 3.3. Modify the make-account procedure so that it creates password-protected accounts. That is, make-account should take a symbol as an additional argument, as in
(define acc (make-account 100 'secret-password))
The resulting account object should process a request only if it is accompanied by the password with which the account was created, and should otherwise return a complaint:
((acc 'secret-password 'withdraw) 40)
60
((acc ‘some-other-password ‘deposit) 50)
“Incorrect password”
Exercise 3.2. In software-testing applications, it is useful to be able to count the number of times a given procedure is called during the course of a computation. Write a procedure make-monitored that takes as input a procedure, f, that itself takes one input. The result returned by make-monitored is a third procedure, say mf, that keeps track of the number of times it has been called by maintaining an internal counter. If the input to mf is the special symbol how-many-calls?, then mf returns the value of the counter. If the input is the special symbol reset-count, then mf resets the counter to zero. For any other input, mf returns the result of calling f on that input and increments the counter. For instance, we could make a monitored version of the sqrt procedure:
(define s (make-monitored sqrt))
(s 100)
10
(s ‘how-many-calls?)
1
Exercise 3.1. An accumulator is a procedure that is called repeatedly with a single numeric argument and accumulates its arguments into a sum. Each time it is called, it returns the currently accumulated sum. Write a procedure make-accumulator that generates accumulators, each maintaining an independent sum. The input to make-accumulator should specify the initial value of the sum; for example
(define A (make-accumulator 5))
(A 10)
15
(A 10)
25
 |
The “eight-queens puzzle” asks how to place eight queens on a chessboard so that no queen is in check from any other (i.e., no two queens are in the same row, column, or diagonal). One possible solution is shown in figure 2.8. One way to solve the puzzle is to work across the board, placing a queen in each column. Once we have placed k – 1 queens, we must place the kth queen in a position where it does not check any of the queens already on the board. We can formulate this approach recursively: Assume that we have already generated the sequence of all possible ways to place k – 1 queens in the first k – 1 columns of the board. For each of these ways, generate an extended set of positions by placing a queen in each row of the kth column. Now filter these, keeping only the positions for which the queen in the kth column is safe with respect to the other queens. This produces the sequence of all ways to place k queens in the first k columns. By continuing this process, we will produce not only one solution, but all solutions to the puzzle.
We implement this solution as a procedure queens, which returns a sequence of all solutions to the problem of placing n queens on an n× n chessboard. Queens has an internal procedure queen-cols that returns the sequence of all ways to place queens in the first k columns of the board.
(define (queens board-size)
(define (queen-cols k)
(if (= k 0)
(list empty-board)
(filter
(lambda (positions) (safe? k positions))
(flatmap
(lambda (rest-of-queens)
(map (lambda (new-row)
(adjoin-position new-row k rest-of-queens))
(enumerate-interval 1 board-size)))
(queen-cols (- k 1))))))
(queen-cols board-size))
In this procedure rest-of-queens is a way to place k – 1 queens in the first k – 1 columns, and new-row is a proposed row in which to place the queen for the kth column. Complete the program by implementing the representation for sets of board positions, including the procedure adjoin-position, which adjoins a new row-column position to a set of positions, and empty-board, which represents an empty set of positions. You must also write the procedure safe?, which determines for a set of positions, whether the queen in the kth column is safe with respect to the others. (Note that we need only check whether the new queen is safe — the other queens are already guaranteed safe with respect to each other.)