# Problem 15:
# 
# [Euler Project #15](https://projecteuler.net/problem=15)
# 
# Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.
# 
# How many such routes are there through a 20×20 grid?
# 
# ---

# Let's try to map the example cases, by assigning a binary value to RIGHT and DOWN moves.

# 0 = R
# 1 = D
#
# 0 0 1 1
# 0 1 0 1 
# 0 1 1 0
# 1 0 0 1
# 1 0 1 0
# 1 1 0 0

# A couple things that shake out of this,
# 1. The total length of a sequence of moves is the equal to Length + Width of the grid.
# 2. Each unique sequence of moves has a twin which is a mirror across the diagonal.
# 3. Each valid sequence has an equal number of RIGHT and DOWN moves.

# I'm sure there is some combinatorial mathematics that describes how to do this analytically,
# but I'd rather practice programming a loop, than researching an elegant solution. 

length=20
width=20
move_sequence_length=length+width

count=0