The crux of the matter is this:
In any triangle, a side of shortest length must be longer
than the difference of the other two sides.
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If a shortest side is 1cm, the other two sides must total 11cm.
But they must differ by less than 1cm, this cannot be, because
if they were 6cm and 5cm, they would differ by 1cm, not less that 1cm.
If a shortest side is 2cm, the other two sides must total 10cm.
But they must differ by less than 2cm, so they can only be 5cm and 5cm
One possible triangle: 2cm, 5cm, 5cm. (An isosceles triangle)
If a shortest side is 3, the other two sides must total 9
But they must differ by less than 3, so they can only be 4 and 5.
A second possible triangle: 3cm, 4cm, 5cm. (A right triangle)
If a shortest side is 4, the other two sides must total 8
But they must differ by less than 4, so they can only be 4 and 4.
The third possible triangle: 4cm, 4cm, 4cm. (An equilateral triangle).
So there are only these 3 possible triangles with an integral number
of centimeters for sides.
Edwin
