Showing what's behind the curtain

Count the ways a row measuring fifty units in length could be filled with blocks three units long. The blocks must be separated by at least one empty space.

Fill an empty row with blocks, a minimum length of 50 units, until the size of the row exceeds 1,000,000 units. The blocks must be separated by at least one empty space.

Count the number of ways a space 50 units long could be filled by tiles, homogeneously, measuring 2, 3 or 4 units long.

Count the number of ways a space 50 units long could be filled by tiles, heterogeneously, measuring 2, 3 or 4 units long.

Prime Subset Sums

Find the number of non-empty subsets of {1^{1}, 2^{2}, 3^{3},…, 250250^{250250}}, the sum of whose elements is divisible by 250.

Maximum sum from top to bottom of triangular array.

Investigating combinations of English currency denominations.

Find the maximum total from top to bottom in a large triangle of numbers using an efficient algorithm.

How many different ways can one hundred be written as a sum of at least two positive integers?