# Project Euler/11

### From charlesreid1

Project Euler problem 11: https://projecteuler.net/problem=11

Solution: https://charlesreid1.com:3000/cs/euler/src/master/011

In this problem you are presented with a 20 x 20 grid, and asked to find the maximum product of 4 sequential integers (up, down, left, right, diagonal up/diagonal down).

## The data structure

The way that I thought about this problem was, we were applying a stencil - a moving window, that was looking at a similar sequence of 4 integers, but applied to different parts of the square. Initially, it was my intention to solve this problem using some kind of recursive backtracking - start at the end and work backwards, trying 4-number combinations as I went. However, this is a gross overcomplication. The stencil idea turns out to be a good one to focus on.

## The procedure

To apply a stencil, you can use a sub-array, and walk the sub-array along the main array structure. Copy operations can also be sped up (if talking about primitive data). In this case, the stencil is a 4 x 4 sub-array. We are looking at valid 4-integer products that are covered by the stencil (specifically, we are computing them, and comparing them to a running maximum to see if we have a new 4-integer sequence yielding a larger product).

The moving stencil operation can also be adapted to linked or array structures, and a great deal more optimization can be done to minimize the number of products computed and operations performed.

## The more general principle

The solution to this problem is one that applies to any number of other problems involving finding combinations of local values - if you build a stencil structure, and shift the structure as you move through all of your data, you can implement the calculations once and keep the task simple.

Note that the way that I implemented it was inefficient - I was re-computing many of the combinations of local values that I had computed before. With greater care and more time this could be improved, but the algorithm is quite cheap for the given problem so speed was not an issue.

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