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Mike Schwörer
2018-02-03 16:50:58 +01:00
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www/statics/euler/Euler_Problem-053_explanation.md
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*That* is the kind of problem I really like. There is a lot of depth in the solution and a proper algorithm solves this in no time.
The algorithm idea here is that we use the properties of [Pascal's Triangle](http://en.wikipedia.org/wiki/Pascal%27s_triangle).
The algorithm idea here is that we use the properties of [Pascal's Triangle](https://en.wikipedia.org/wiki/Pascal%27s_triangle).
As you probably know this triangle starts at its top with a `1`. Then every cell is calculated as the sum of the two cells above it.
~~~
cell[y][x] = cell[y-1][x-1] + cell[y-1][x]
~~~
![Animated GIF of Pascals Triangle. (c) by Wikimedia Foundation](http://upload.wikimedia.org/wikipedia/commons/0/0d/PascalTriangleAnimated2.gif)
![Animated GIF of Pascals Triangle. (c) by Wikimedia Foundation](https://upload.wikimedia.org/wikipedia/commons/0/0d/PascalTriangleAnimated2.gif)
When we calculate `C(n, r)` this is nothing more than the number at row *n* and column *r*.
So what we do is build [Pascal's Triangle](http://en.wikipedia.org/wiki/Pascal%27s_triangle) up to a height of one-hundred and look at the cells with values greater than one million.
So what we do is build [Pascal's Triangle](https://en.wikipedia.org/wiki/Pascal%27s_triangle) up to a height of one-hundred and look at the cells with values greater than one million.
The obvious problem is that the numbers grow extremely big, and sooner or later *(probably sooner)* the numbers will overflow. So what we do is use a little trick:
As soon as a cell is over `1 000 000` we mark her *(= put a zero in that cell)*. When we create a new cell out of its two parents we check if one of the parents has the mark *(= is zero)* and then the children gets also marked. Because if one of the parents (or both) is over one million then all of its children will also be over one million.