Clojure Euler: Problem 015

Starting in the top left corner of a 2 x 2 grid, there are 6 routes (without backtracking) to the bottom right corner.

How many routes are there through a 20 x 20 grid?

Permalink: http://projecteuler.net/problem=15

Picture is always a huge help to find correct the solution. Let's see the pattern here, with another picture

Warning: If you don't understand art, don't look at that picture.

It's a solution for 3 x 3 grid. We trying to process it by steps:

If you just output the numbers from each diagonal line you'll get:

       1
      1 1
     1 2 1
    1 3 3 1

It's famous Pascal's triangle. Each element in this triangle obtained by sum of two closest elements from the upper row.

To get desired result we need to sum these numbers again, until we get one number. But it's also numbers from Pascal's triangle.

    1   3   3   1
      4   6   4
       10  10
         20

We can treat Pascal's triangle as a sequence, and each row is an element. Then the following function calculates next row:

(defn routes-extend [lst]
  (let [size (count lst)]
    (for [i (range (inc size))]
      (if (or (= 0 i) (= size i)) 1
        (+ (nth lst (dec i)) (nth lst i))))))

Usage:

(routes-extend [1]) => [1 1]
(routes-extend [1 1]) => [1 2 1]

Now we can build infinite sequence of Pascal's triangle rows:

(iterate routes-extend [1])

For 3 x 3 grid we had 7 diagonals, each of them represented one row. Take a guess. For n x n grid we need to get 2 * n + 1 row. And take it middle element, which, of course, will be with the index n.

That's the final solution:

(let [n 20 d (inc (* n 2))]
  (nth (last (take d (iterate routes-extend [1]))) n))

Problem solved.

Code

P.S Two points I want to admit here. First of all, drawing pictures and trying to understand problem with pictures is a powerful technique that must be used for every problem.

Second of all, you can see that our algorithm has complexity of O(n^2). But ideal solution is O(1) with usage binomial coefficient (2*n, n).

mishadoff 19 February 2013
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