CS 307: 2. Playing Peano and Gambling

Due: Wednesday, September 12, 2001.

  1. Write a function (sum-squares n) that will recursively add up the squares of the integers from 1 to n.
  2. The mathematician Giuseppe Peano (1858 - 1932) showed that all arithmetic operations on natural numbers (nonnegative integers) could be reduced to a single constant (0) and an operation successor, which in Lisp traditionally is called 1+ .
    ``Peano was one of the first to use what we now call symbolic logic. He introduced, for instance, the use of the symbols `(E x)' to mean `there is an x such that'; and he habitually wrote out all of his lecture notes in his new symbolism. He was teaching at a military academy at the time, and his students were so incensed by his formalistic approach to mathematics that they rebelled (despite his promises to pass them all) and got him fired. Subsequently he found a more congenial setting at the University of Turin.'' -- Rudy Rucker, Goedel's Incompleteness Theorems, p. 289.

    The functions (1+ n) and (1- n), which add and subtract 1 from an integer, are provided in the file initdr.scm

    Write a function (peano+ x y), using only 1+ and 1- to perform arithmetic, according to the following definition:
    peano+(x, y) = x , if y = 0
    peano+(x + 1, y - 1) , otherwise.
    Can you think of an invariant (property that is always true) of peano+? (Hint: trace the function and run it on an example.) This function is tail-recursive.

  3. Write a function (peano* x y) that uses only peano+, 1+, and 1- to perform arithmetic and is written in a recursive style similar to that of peano+. For peano*, use an auxiliary function and an auxiliary variable to accumulate the result so that it will be tail-recursive.

  4. Test your functions on several sets of positive integer arguments to verify that they perform addition and multiplication.

  5. The mathematical notation , read ``n choose k'', is used to denote the number of distinct subsets of k items that can be chosen from a set of n distinct objects. It can be shown that:

    Although the factorial function could be used in implementing n choose k, this would be inefficient for large values of n and small k. We can algebraically rewrite the definition into the following form for k > 0:

    Write a function (choose n k), using a tail-recursive auxiliary function, to compute n choose k without using the factorial function. Can you prove that the result of your auxiliary function is always an integer? This would be a good invariant for your recursive function: although non-integer intermediate results will work in Lisp, they would not work in most languages.

  6. The Texas lottery involves a choice of 6 numbers from a set of 54. Use your function to compute 54 choose 6 to determine the probability of winning the jackpot.

  7. If you buy 20 lottery chances (for $20) every week for the next 50 years, what is the probability that you will win the jackpot? Assuming that the jackpot averages $8 million and that the jackpot is the only prize, what is your expected return (probability of winning times amount won)?

  8. Write a recursive function (invest amt time int) to calculate the return on an investment of amount amt for time time periods at an interest rate of int. Assume that the starting principal is 0. At the end of each time step, the value of the principal becomes the value of the original principal times (1 + int), where int is the interest rate per time period, plus the amount amt of the new contribution.

  9. If instead of buying lottery tickets, you invest $20 per week for 50 years at an interest rate of 6.0% per year, what is the value of your investment at the end of 50 years? You may do the calculation on a yearly basis, i.e., $1040 per year as the contribution per year.