CIT 591 Assignment 2: Figurate Numbers
Fall 2014, David Matuszek

Purposes of this assignment

General idea of the assignment

In this assignment you will write a number of functions for testing characteristics of positive integers, plus one special function named main. Each function other than main will take any single positive integer as an argument, test whether it has some property, and return a value of either True or False..

Put your program on a file named figurate_numbers.py. Get a copy of the file figurate_numbers_test.py. (As you read this, the file may be missing or incomplete; if so, it will be ready soon.)

Details

The main() function

In the main function, define a variable limit = 100 and test each of the numbers 1 through limit, inclusive. Do not use the number 100 elsewhere; use the variable limit instead. This makes the program easier to change if you later want some limit other than 100.

The main function will call each of the other functions, for each of the numbers 1 through limit, to determine the properties of those numbers. It will then print out each number, one per line, along with a list of its properties (on the same line).

Your output should look approximately like this:

    1 composite, triangular, tetrahedral, square,  square pyramidal, pentagonal, not prime oblong, not pointy
    2 prime, not triangular, not tetrahedral, not square,  not square pyramidal, not pentagonal, not prime oblong, not pointy
    3 prime, triangular, not tetrahedral, not square,  not square pyramidal, not pentagonal, not prime oblong, not pointy
    4 composite, not triangular, tetrahedral, square,  not square pyramidal, not pentagonal, not prime oblong, not pointy
    5 prime, not triangular, not tetrahedral, not square,  square pyramidal, pentagonal, not prime oblong, pointy
    . . .

The main function doesn't return a value. Each of the other functions is a predicate, that is, a function that returns True or False.

The is_prime(n) function

A positive number is prime if its only positive divisors are itself and one. For example, 7 is prime because it is evenly divisible by 1 and 7, but not by any number in between (in particular, it is not evenly divisible by 2, 3, 4, 5, or 6).

A positive number is composite if it is not prime. For example, 10 is composite because it is evenly divisible by 2 and 5; 12 is composite because it is evenly divisible by 2, 3, 4, and 6. As a special case, 1 is considered to be composite.

This function should return True if its argument is a prime number, and False otherwise. Similar statements hold for the other functions.

The is_triangular(n) function

A triangular number is a number of the form 1 + 2 + 3 + ... + n, for some positive n. These numbers are called triangular because, if you have that many objects, you can arrange them in an equilateral triangle (see figure).

The first few triangular numbers are 1, 3, 6, 10, 15....

stacks of balls

The is_tetrahedral(n) function

To build tetrahedral numbers, 1, 4, 10, 20, 35, 56, etc., we successively stack triangular numbers, 1, 3, 6, 10, 15, ..., one at
a time.

tetrahedral

The is_square(n) function

A square number is a number of the form 1 + 3 + 5 + 7 + ... + n, for some odd positive n. (The figure should help you understand why this definition is the same as the usual definition of square numbers.)
Balls in a square

The is_square_pyramidal(n) function

To build square pyramidal numbers, 1, 5, 14, 30, 55, 91, etc., we successively stack the square numbers, 1, 4, 9, 16, 25,36, etc., one at a time.

square pyramidal

The is_pentagonal(n) function

Pentagonal numbers can be generated by triangular ones, as seen here to the left, yielding the general formula
Pn= 1 + 4 + 7+... + (3n-2) = Tn+2Tn-1= ½n(n +1) + n(n - 1) = ½n (3n - 1).

pentagonal

The is_prime_oblong(n) function

(My invention, not a standard mathematical term). Prime oblong numbers are those that have exactly two different prime divisors. Thus, 10 (2 * 5) is prime oblong, 12 is not (too many divisors), 25 is not (two equal prime divisors).

 

The is_pointy(n) function

A number is pointy (again, my invention, not a standard mathematical term) if it is the sum of two different triangular tetrahedral numbers. The first few pointy numbers are 5 (1+4), 11 (1+10), 14 (4+10), 21 (1+9), 24 (4+20), 30 (10+20), etc.

Coda

At the end of your Python program (after all your function definitions), insert the following lines:

if __name__ == "__main__":
  main()

Here's what this does. If you are in the IDLE window containing your program, and you click F5 (or choose Run -> Run module) from the menu, IDLE will automatically run your main function. If you are in the IDLE window containing your test cases (supplied) and click F5 or use the menu equivalent, it will run the tests and tell you the results.

References

The figures and descriptions of tetrahedral, square pyramidal, and pentagonal numbers are from Mathematics: From the Birth of Numbers by Jan Gullberg.

Due date

This is a pair programming project. Only one of you, either you or your team member, should turn in the assignment. Make sure both your names are in comments at the top of the file.

Turn in your assignment by 6am, Friday September 12 . Turn in only the Python file you wrote; don't turn in the supplied test file. As only one file is required, there is no need to zip it.