What is the maximum number of triangles that can be formed using 87 sticks of…

Key idea: place the triangles in a row so each new triangle shares one side with the previous triangle.

• The first triangle requires 3 sticks.

• Each additional triangle (sharing one side) requires 2 more sticks.

• If T is the number of triangles, total sticks needed = 3 + 2*(T - 1).

Solve the inequality 3 + 2*(T - 1) ≤ 87 to find the maximum T.

1. 3 + 2(T - 1) ≤ 87

2. 2T + 1 ≤ 87 ⇒ 2T ≤ 86 ⇒ T ≤ 43

Therefore the maximum number of triangles is 43. Check: 3 + 2*(43 - 1) = 3 + 84 = 87, which uses all sticks.

Note: If sticks are not shared at all, you would get ⌊87/3⌋ = 29 triangles, but sharing sides allows more triangles, giving the optimal 43.