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ParsedownCustom

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2018-01-01 23:03:05 +01:00
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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.
![Grid Image](/data/blog/Befunge/p015.gif)
![Grid Image](/data/images/blog/p015.gif)
How many such routes are there through a 20×20 grid?

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Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
![](https://projecteuler.net/project/images/p068_1.gif)
![](/data/images/blog/p068_1.gif)
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely.
For example, the above solution can be described by the set: `4,3,2; 6,2,1; 5,1,3`.
@@ -23,4 +23,4 @@ By concatenating each group it is possible to form 9-digit strings; the maximum
Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings.
What is the maximum 16-digit string for a "magic" 5-gon ring?
![](https://projecteuler.net/project/images/p068_2.gif)
![](/data/images/blog/p068_2.gif)

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By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles:
![Grid Image](/data/blog/Befunge/p085.gif)
![Grid Image](/data/images/blog/p085.gif)
Although there exists no rectangular grid that contains exactly two million rectangles,
find the area of the grid with the nearest solution.

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The points `P(x1, y1)` and `Q(x2, y2)` are plotted at integer co-ordinates and are joined to the origin,
`O(0,0)`, to form `OPQ`.
![img](/data/blog/Befunge/p091_1.gif)
![img](/data/images/blog/p091_1.gif)
There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate
lies between 0 and 2 inclusive; that is,
@@ -10,6 +10,6 @@ lies between 0 and 2 inclusive; that is,
0 <= x1, y1, x2, y2 <= 2.
~~~
![img](/data/blog/Befunge/p091_2.gif)
![img](/data/images/blog/p091_2.gif)
Given that `0 <= x1, y1, x2, y2 <= 50`, how many right triangles can be formed?