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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Half solution -- the second half | Comment 5 of 6 |
(In reply to re: Half solution -- the second half by Charlie)

 
Using Pick's Theorem: Area = I + B/2 - 1
Points in the interior
  I
add one-half for points on border (not vertices)
     + (B - 3)/2
add one-half for vertices
                  + 1/2
gives
  I + (B - 3)/2 + 1/2 =
  [Area - B/2 + 1] + (B - 3)/2 + 1/2 =
  Area =
  x*y/2
 

Edited on June 9, 2005, 9:43 pm

Edited on June 15, 2005, 2:42 am
  Posted by Bractals on 2005-06-09 21:17:31

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