Euler Problem 15: Pathways Through a Lattice

March 9, 2017

(This article was first published on The Devil is in the Data, and kindly contributed to R-bloggers)

Euler Problem 15 analyses taxicab geometry. This system replaces the usual distance function with the sum of the absolute differences of their Cartesian coordinates. In other words, the distance a taxi would travel in a grid plan. The fifteenth Euler problem asks to determine the number of possible routes a taxi can take in a city of a certain size.

Euler Problem 15 Definition

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 possible routes are there through a 20×20 grid?

Euler Problem 15: Lattice Paths


The defined lattice is one larger than the number of squares. Along the edges of the matrix, only one pathway is possible: straight to the right or down. We can calculate the number of possible pathways for the remaining number by adding the number to the right and below the point.


For the two by two lattice the solution space is:

6  3  1
3  2  1
1  1  0

The total number of pathways from the upper left corner to the lower right corner is thus 6. This logic can now be applied to a grid of any arbitrary size using the following code.

The code defines the lattice and initiates the boundary conditions. The bottom row and the right column are filled with 1 as there is only one solution from these points. The code then calculates the pathways by working backwards through the matrix. The final solution is the number is the first cell.

# Define lattice
nLattice <- 20
lattice = matrix(ncol=nLattice + 1, nrow=nLattice + 1)

# Boundary conditions
lattice[nLattice + 1,-(nLattice + 1)] <- 1
lattice[-(nLattice + 1), nLattice + 1] <- 1

# Calculate Pathways
for (i in nLattice:1) {
    for (j in nLattice:1) {
        lattice[i,j] <- lattice[i+1, j] + lattice[i, j+1]

answer <- lattice[1,1]

Taxicab Geometry

The post Euler Problem 15: Pathways Through a Lattice appeared first on The Devil is in the Data.

To leave a comment for the author, please follow the link and comment on their blog: The Devil is in the Data. offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.


Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)