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Dotty Right Triangle (Posted on 2005-06-09) Difficulty: 3 of 5

A "dot" (commonly also called a "lattice point") is a point with integer coordinates.

In the plane, what is the total number of dots inside or on the boundary of the triangle with vertices (0,0), (x,0), (x,y) where x and y are positive integers?

In the event that it is not utterly obvious from the form of your answer that a whole number is being specified, give an independent argument to show this.

What total do you get if you count the three vertex dots together as just half a dot and any other boundary dots as half a dot each?

See The Solution Submitted by Richard    
Rating: 4.5000 (2 votes)

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Solution re: Half solution -- the second half | Comment 3 of 6 |
(In reply to Half solution by Bractals)

Since

[ (x+1)*(y+1) + gcd(x,y) + 1 ] / 2

represents the total number of points within or on the edge of the triangle, and the second parts asks for those points at the vertices to count only as 1 altogether, and those points elsewhere on the edge to count as 1/2,we just have to subtract out:

2 (for the three vertices counting as one)
(y-1)/2 (halving along the vertical edge, other than the vertices)
(x-1)/2 (halving along the horizontal edge, other than the vertices)
(gcd(x,y)-1)/2 (halving along the diagonal, NOT counting the two vertices)

So it comes out to

[ (x+1)*(y+1) - x - y ] / 2


  Posted by Charlie on 2005-06-09 19:36:35
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