Brick Tower Building Puzzle

Question

I\'ve been given a little brainteaser to solve.

The task is to make a function with a single integer parameter. You have to figure out how many different combination

Answer 1

Wikipedia gives the generating function for the number of partitions of n with distinct parts as q(n) = product (1+x^k) for k=1..infinity. Given that you exclude the possibility of a single tower, the number of different valid tower arrangements is q(n)-1.

This gives this neat O(n^2) time and O(n) space program for counting tower arrangements.

def towers(n):
    A = [1] + [0] * n
    for k in xrange(1, n+1):
        for i in xrange(n, k-1, -1):
            A[i] += A[i-k]
    return A[n] - 1

print towers(200)

The output is as required:

487067745

To understand the code, one can observe that A stores the first n+1 coefficients of the generating function product(1+x^k) for k=1...infinity. Each time through the k loop we add one more term to the product. We can stop at n rather than infinity, because subsequent terms of the product do not affect the first n+1 coefficients.

Another, more direct, way to think about the code is to define T(i, k) to be the number of tower combinations (including the single tower) with i blocks, and where the maximum height of any tower is k. Then:

T(0, 0) = 1
T(i, 0) = 0 if i > 0

T(i, k) = T(i, k-1)               if i < k
        = T(i, k-1) + T(i-k, k-1) if i >= k

Then one can observe that after j iterations of the for k loop, A contains the values of T(j, i) for i from 0 to n. The update is done somewhat carefully, updating the array from the end backwards so that results are changed only after they are used.