Math CentralQuandaries & Queries


Question from Saurabh, a student:

Number of points with integer coordinates which lie inside the triangle whose vertices are (0,0) , (0,21) and (21,0) ???

Hi Saurabh.

Let me try a similar triangle: (0,0), (10,0), (0,10).

I assume that points on the side of the triangle are not "inside" it, so looking at each vertical line that is an integer value of x, I have x = 1, where the inside points go from y = 1 up to y = 8. We don't include y=9 because it is on the line itself, not inside.

The next value of x is x=2 where the possible integer values of y are y=1 up to y=7.

So you can see that we can just add up the numbers:
8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
to answer the question.

Another way to finish the problem is to use a "summation formula". To add all the natural numbers up to n, we just calculate n(n+1) / 2. So for 1+...+8 we have 8(8+1)/2 = 36.

Now you try your question using this model.

Stephen La Rocque.

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