3. If you continue the chain of squares and regular pentagons in the same way, does it close precisely upon itself after going all the way around? If so, how many pentagons does the chain contain in total?
A) 9
B) 10
C) 11
D) 12
E) It does not close.

![](https://cdn.mathpix.com/cropped/2024_04_17_51d50f3426d3742e694cg-1.jpg?height=411&width=374&top_left_y=2187&top_left_x=1521)
A3. B) 10 The angles of a regular $n$-gon are equal to $180-\frac{360}{n}$ degrees. Indeed, if you walk around the $n$-gon, you change direction $n$ times, after which you have turned a total of 360 degrees. At each vertex, your direction changes by $\frac{360}{n}$ degrees, and the corresponding angle of the $n$-gon is equal to $180-\frac{360}{n}$ degrees.

The angles of a square are thus $180-90=90$ degrees, and those of a regular pentagon are $180-72=108$ degrees.

If we place a square and a regular pentagon against each other, as shown in the figure, they together enclose an angle of $360-90-108=162$ degrees (angle $A B C$ in the figure). Since $162=180-18$, this is precisely the angle of a regular 20-gon because $18=\frac{360}{20}$. The sides $A B$ and $B C$ are of equal length, so after twenty steps (10 squares and 10 pentagons), we are exactly back where we started, and the inner sides of the squares and pentagons form the sides of a regular 20-gon.

![](https://cdn.mathpix.com/cropped/2024_04_17_89d61e9f56efbfe4ed42g-1.jpg?height=408&width=717&top_left_y=2309&top_left_x=681)