Numerical answer
A right triangle has its right-angle vertex at the origin and its two legs along the axes, of lengths 18 units each. How many points with both coordinates integers lie strictly inside this triangle?
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Answer: 136
The triangle has vertices (0, 0), (18, 0) and (0, 18). Interior points need x ≥ 1, y ≥ 1 and x + y < 18, i.e. x + y ≤ 17.
For x = 1, y can be 1 to 16 (16 values); for x = 2, 15 values; … for x = 16, 1 value.
Total = 16 + 15 + … + 1 = 136.
Check using Pick's theorem: Area = 162, boundary points B = 54 (18 on each leg, 18 on the hypotenuse since gcd(18,18)=18, sharing the 3 vertices). A = I + B/2 − 1 → 162 = I + 27 − 1 → I = 136. ✓
For x = 1, y can be 1 to 16 (16 values); for x = 2, 15 values; … for x = 16, 1 value.
Total = 16 + 15 + … + 1 = 136.
Check using Pick's theorem: Area = 162, boundary points B = 54 (18 on each leg, 18 on the hypotenuse since gcd(18,18)=18, sharing the 3 vertices). A = I + B/2 − 1 → 162 = I + 27 − 1 → I = 136. ✓
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