The perimeter of a triangle is 12 cm. If all the three sides have lengths (in cm), in integers, then how many such different triangles are possible?

We need integer sides a ≤ b ≤ c with a+b+c = 12 and a+b > c (triangle inequality). Try c values: c=4: a+b=8, a≤b≤4 gives (4,4,4). c=5: a+b=7, b≤5 gives (3,4,5) since 3+4>5. c=6: a+b=6, b≤6 and a+b>6 fails (6=6). Wait, b≤5 since a+b>c. So (2,4,6) fails, (3,3,6) fails (3+3=6 not >6). c=5: (2,5,5) works since 2+5>5. So valid: (4,4,4), (3,4,5), (2,5,5). That's 3 triangles.