The game "Geographic Dice" consists of each participant rolling a die a number of times $n$ and recording the sequence, where the first roll indicates the distance, in meters, that the participant will walk to the North, the second to the East, the third to the South, the fourth to the West, the fifth to the North, and so on. After that, the distance to the origin is measured. The winner is the one who ends up closest to the starting position.

a) Márcia rolled 214365 (6 rolls). What is her distance from the origin?
b) In 4 rolls, how many possibilities exist for the participant to return to the origin after the last roll?

c) In 5 rolls, how many possibilities exist for the participant to return to the origin after the last roll?
Solution

a) Márcia walked $2+6=8 \mathrm{~m}$ to the North, $1+5=6 \mathrm{~m}$ to the East, $4 \mathrm{~m}$ to the South, and $3 \mathrm{~m}$ to the West. In the North-South direction, Márcia walked 8-4 = $4 \mathrm{~m}$ to the North, and in the East-West direction, Márcia walked $6-3=3 \mathrm{~m}$ to the East. Since these directions are orthogonal, we can apply the Pythagorean Theorem to determine the distance $d$ from the origin: $d^{2}=4^{2}+3^{2}$, so $d=5 \mathrm{~m}$.

b) To return to the origin, the participant must roll the same result for North and South, and also the same for East and West. Since there are 6 possible equal values in two rolls, the total number of possibilities to return to the origin at the end of the last roll is $6 \cdot 6=36$.

c) We will use the same reasoning as in the previous item, but there will be 2 dice determining the displacement to the North (first and fifth). For the East-West direction, there are 6 possibilities. If we roll a 1 for South, there is no possibility of returning to the origin with 2 rolls for the North; if we roll a 2 for South, there is only 1 possibility for the North $(1+1)$; if we roll a 3 for South, there are 2 possibilities for the North ( $1+2$ or $2+1$ ); if we roll a 4, there are 3 possibilities; if we roll a 5, there are 4 possibilities; if we roll a 6, there are 5 possibilities. Thus, there are $1+2+3+4+5=15$ possibilities in the North-South direction. Therefore, there are $6 \cdot 15=90$ possibilities of returning to the origin after 5 rolls.