????

i give

How many triangles and quadrilaterals are in this diagram?

There is a pattern for the triangles:

if n the number of the lines in the triangle on every side, then the number of triangles is \((n+1)^3\)

We have n = 3, so \((n+1)^3 = (3+1)^3 = 4^3 = 64\) triangles.

I assume the pattern for the quadrilaterals is:

\(n=0: \quad (0)^2 = 0 \\ n=1: \quad (0+1)^2 =1^2 = 1 \\ n=2: \quad (0+1+2)^2 = 3^2 = 9 \\ n=3: \quad (0+1+2+3)^2 = 6^2 = 36 \\ \dots \\ \text{for n}: \qquad \left[\frac{(n+1)\cdot n}{2} \right]^2 \text{quadrilaterals} \)

We have n = 3, so \( \left[\frac{(n+1)\cdot n}{2} \right]^2 = \left[\frac{(3+1)\cdot 3}{2} \right]^2=\left[\frac{4\cdot 3}{2} \right]^2=6^2=36\) quadrilaterals