Project Euler 115: Fill an empty row with blocks, a minimum length of 50 units, until the size of the row exceeds 1,000,000 units.

Problem Description

NOTE: This is a more difficult version of Problem 114.

A row measuring n units in length has red blocks with a minimum length of m units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.

Let the fill-count function, F(m, n), represent the number of ways that a row can be filled.

For example, F(3, 29) = 673135 and F(3, 30) = 1089155.

That is, for m = 3, it can be seen that n = 30 is the smallest value for which the fill-count function first exceeds one million.

In the same way, for m = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so n = 57 is the least value for which the fill-count function first exceeds one million.

For m = 50, find the least value of n for which the fill-count function first exceeds one million.

Analysis

This problem could easily be solved by playing with the interactive solution on the Problem 114 page. Just change m to 50 and play with n until it first exceeeds one million. The value of n won’t be very big because we start m at 50.

Again, i opted out of a recursive solution or generating functions because, after a while, dynamic programming becomes so intuitive for solving counting problems.

Also, I didn’t bother with an F() function as I wanted to simply make a few changes to the previous problem and finish it. Instead of a fixed-size array, I dynamically increase it as necessary since I have no clue as to how big it should be to start with.

Well, actually, I do have a clue, but not if the parameters of this problem were changed.

Project Euler 115 Solution

Afterthoughts

• See also, Project Euler 114 Solution: