**What is Project Euler?**

Project Euler is a series of challenging problems that require mathematical and programming skills. Somebody who enjoys learning new area of mathematics, project Euler is going to be a fun journey.

**Where are the problems ?**

The problems are right here in their official archive.

Let’s solve a problem from the archive and understand its complexity. Randomly I have chosen Problem no 116.

**Problem 116 : Red, green or blue tiles**

Problem Statement

**Solution: [ IT IS ADVISED TO TRY YOURSELF FIRST]**

A red tile is of length 2, green is of length 3 and blue is of length 4.

Since, we need to count total ways for 50 units of black colored square tiles, say **k = 50**.

def E_116(i, k): ways = [1] * i + [0] * (k-i+1) for j in range(i, k+1): ways[j] += ways[j - 1] + ways[j - i] return ways[k] - 1

Here, we are initializing our function **E_116()** which holds the logic of the solution to the problem.The function **E_116()** has two parameters **i** = number of black coloured square tiles covered by the new coloured (red, green or blue) tiles and **k** = total number of black coloured square tiles.

In the function,

ways = [1] * i + [0] * (k-i+1)

So, ways is a list which holds the total number of ways which the i length block can cover 50 black coloured square tiles.For e.g:

In the above example x = [1, 1, 1, 0, 0, 0] , has 1 (x3) and 0 (x3) , where 1 represents possible solution case and 0 represents failure.As we can compare for i = 3 and k = 5 from the question, we get total 3 possible ways.Hence there are 1 (x3) in the list, ways.

for j in range(i, k+1): ways[j] += ways[j - 1] + ways[j - i]

Using the for loop we are iterating from i to 50, we have written k+1 since iterating in the loop through range() will exclude the last case.In **ways[j] += ways[j – 1] + ways[j – i]**, we are updating the jth index of the list ways with the summation of (j-1) and (j+i)th index’s value and also the jth index value (Since **+=**).

These are the possible values of j, for i=3 and k =5.

Finally we return our list as **return ways[k] – 1**.So, for i=3 and k=5, this gives the solution(i.e, Ans = 3)

Lastly in our code,

print("Number of black tiles =", k, "units") print("Number of ways to fill:", E_116(2, k) + E_116(3, k) + E_116(4, k))

We print away our result using the print() function. The first statement prints ** Number of black tiles = 50** and the second statement prints

**which is the desired answer to the problem.**

*Number of ways to fill: 20492570929*
`# Project Euler Problem 116 ` ` ` `k ` `=` `50` `def` `E_116(i, k): ` ` ` `ways ` `=` `[` `1` `] ` `*` `i ` `+` `[` `0` `] ` `*` `(k` `-` `i` `+` `1` `) ` ` ` `for` `j ` `in` `range` `(i, k` `+` `1` `): ` ` ` `ways[j] ` `+` `=` `ways[j ` `-` `1` `] ` `+` `ways[j ` `-` `i] ` ` ` `return` `ways[k] ` `-` `1` ` ` `print` `(` `"Number of black tiles ="` `, k, ` `"units"` `) ` `print` `(` `"Number of ways to fill:"` `, E_116(` `2` `, k) ` `+` `E_116(` `3` `, k) ` `+` `E_116(` `4` `, k)) ` |

**Resources:**

This article is contributed by **Amartya Ranjan Saikia**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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