Comp 210 Lab 10: Accumulators, Homework hints

Accumulators, Homework hints

Consider the function f : positive -> positive,

f(n) =	1	if n=1
	n/2	if n is even
	3n+1	if n is odd

While it may look like a pretty mundane function, try iterating it (repeating it over and over, using the previous result), and seeing how long it takes to reach 1 (if ever). This is known as Colatz's function, and we've seen it before in threes:

; threes : N[>=1] -> N[>=1]
(define (threes n)
   (cond [(= n 1)  1]
         [else     (cond [(even? n) (threes (/ n 2))]
                         [else      (threes (+ 1 (* n 3)))])]))

An example: if we start with (say) 5, we find that:

• f(5)=16,
• f(16)=8,
• f(8)=4,
• f(4)=2, and
• f(2)=1.

To do: Write a program stops? which takes a number n, and returns true if n, f(n), f(f(n)), ... ever reaches the number 1, i.e., if (threes n) stops. (What does your program do if this doesn't happen?) It should not call threes, but it will resemble it. (Hint: Add an accumulator to a copy of threes.)

Q: Why do we choose to stop at f(1)?

Q: Does threes stop for all inputs? (Don't feel bad if you can't answer this. No one else has yet.)

To do: Now, write a function time-til-stop, which instead of returning true, it returns the number of steps until 1 is reached. That is, (time-til-stop 5) = 5. Test your program on some large numbers, say 1001, 2349872349247990, and 2349872349247991.

To do: Recall the equational definition of foldl:

     (foldl f base (list x1 x2 ... xN)) = (f xN ... (f x2 (f x1 base))...)

In lab, we will discuss some ideas about what states are for the Missionaries & Cannibals problem. The following are some generalities to consider:

1. The first step is to think about is what pieces of information are important to know. For this problem, what do we need to know about the missionaries, cannibals, boat, and sides of the river?
2. The second step is to choose how to represent that data.
   • Can we place a fixed bound on the amount of data? If so, we probably shouldn't use a recursive datatype, but if not, we must use a recursive datatype.
   • Is the data something common, like a number or a list of numbers? If not, we should probably define a datatype with define-structure so that we use a readable set of constructors, predicates, and selectors.
   • There are frequently several ways to represent a datatype. Which ones allow us to write our program more easily?
   • Are parts of the data logically associated with each other? E.g., a bunch of locations, each with its own temperature? If so, it is generally helpful to use nested data structures, such as a list of pairs, e.g., a list of city/temperature pairs.
   • Is any of the data redundant?

     E.g., if we are representing only right triangles, the data might be the lengths of the three sides, but having two side lengths is sufficient. We may want to keep track of only the minimal amount of information (here, two sides' lengths), or all of the data (here, all three sides' lengths). The latter choice makes it easier to get at all of the data, but we may have to worry about having inconsistent data (here, three lengths that don't make a right triangle).

Your M&C program needs to build up a list of moves to the result. That sounds sort of similar to our in-class example that built up a list of cities for a path in a graph.