How many points with both coordinates integers lie strictly inside the triangle with vertices (0, 0), (10, 0) and (0, 10)?

A point (x, y) is strictly inside if x ≥ 1, y ≥ 1 and x + y ≤ 9 (points with x + y = 10 lie on the hypotenuse).

For x = 1, y can be 1…8 (8 values); x = 2 → 7 values; … ; x = 8 → 1 value.

Total = 8 + 7 + … + 1 = 36.

Check with Pick's theorem: Area = 50, boundary points B = 10 + 10 + 10 = 30. Area = I + B/2 − 1 → 50 = I + 15 − 1 → I = 36 ✓.

45 counts points on the hypotenuse too; 55 includes points on the axes.