Problem 2 · The ring of small triangles
Generalising the pattern: how many unit triangles surround a triangle of side $n$.
Integer answer, at most 4 digitsThe figure shows an equilateral triangle of side 3 cm surrounded by 24 smaller triangles. Another, larger equilateral triangle is surrounded in the same way by 102 small triangles equal to the previous ones. What is the side length of this new triangle?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: count the ring for a generic side $n$ and check the formula against the known case $n=3$.
Along each side of the triangle of side $n$ there is a strip of alternating unit triangles: $n$ pointing outwards and $n+1$ pointing inwards, $2n+1$ per side in total. In addition, each corner holds $1$ extra triangle.
$$T(n) = 3(2n+1) + 3 = 6n + 6.$$
Check against the figure: $T(3) = 6\cdot 3 + 6 = 24$ ✓.
$$6n + 6 = 102 \;\Longrightarrow\; n = 16.$$
Answer: 16