19) Giovanni has drunk too much and starts to walk in a strange way:

- he takes 1 step forward;
- then he turns $90^{\circ}$ to the right and takes 2 steps forward;
- then he turns $90^{\circ}$ to the right and takes 1 step forward;
- then he turns $90^{\circ}$ to the left and takes 1 step backward; - after that, he starts over.

Each step is 1 meter. After 186 steps, he falls to the ground unconscious. How many meters from where he started does Giovanni end his walk?
(A) $0 \quad$ (B) 1
(C) 2
(D) $\sqrt{5}$
(E) 4 .
19) The answer is $(\mathrm{C})$. Let "origin" be Giovanni's initial position, and "North" the direction he is facing. It is noted that after the first 5 steps (see the figure on the side), Giovanni is at the point $(1,0)$ and facing East. Therefore, every 5 steps, Giovanni is moved one step to the right and rotated $90^{\circ}$ to the right. After $4 \cdot 5=20$ steps, Giovanni will be back at the origin, facing North, i.e., in the same initial position. After the first 180 steps (a multiple of 20), Giovanni is still at the starting point, facing North. With another 5 steps, he will be at the point $(1,0)$ and facing East; with one more step forward (the 186th step), he will reach the point $(2,0)$,

![](https://cdn.mathpix.com/cropped/2024_04_17_8c48e706878e314feb6bg-4.jpg?height=603&width=793&top_left_y=1526&top_left_x=1134)
two meters from the starting point.