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 perplexus dot info

 Dotty Right Triangle (Posted on 2005-06-09)

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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 Half solution | Comment 2 of 6 |
` `
`The number of points on the diagonalis the gcd(x,y)+1 (this is countingthe two vertices).`
`If we look at the number of pointsin or on the corresponding rectangle `
`  (0,0) (x,0) (x,y) (0,y)`
`it is (x+1)*(y+1).`
`If we subtract the number of pointson the diagonal, divide by two, andadd back the points on the diagonalwe get the number of points in or onthe triangle:`
`  [ (x+1)*(y+1) + gcd(x,y) + 1 ] / 2`
`   x     y    (x+1)*(y+1)  gcd(x,y)`
`  even  even      odd        even  even  odd       even       odd  odd   even      even       odd  odd   odd       even       odd`
`The numerator in all cases is even.`
`Have not looked at the second part yet.`
` `

 Posted by Bractals on 2005-06-09 18:43:12

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