A regular icosahedron is a geometric solid with 20 faces, which are equilateral triangles. By drawing the numbers from 1 to 20 on the faces of a regular icosahedron, it is transformed into a die. Luísa and Mathias invented a game in which each of them rolls the icosahedral die 5 times and records the numbers drawn in sequence, forming a single number. For example, Luísa rolled the sequence $5,11,7,20,17$, so she wrote down the number 51,172,017.

a) In the first game, Luísa won with the largest possible difference. What is this difference?

b) In the second game, Luísa scored 162,012,510 points and Mathias rolled the die four times, drawing 14, 11, 8, and 19. Is it possible for Mathias to win this game with his last roll?

c) In the third game, Luísa scored 12,111,813 points. How many sequences are possible to achieve this score?
Solution

a) 2,020,190,909. For this to happen, Luísa scored 20 in her 5 throws and Mathias scored 1 in his 5 throws, making the difference equal to 2,020,202,020 - 11,111 = 2,020,190,909.

b) No. The highest possible score would be by scoring 20 on the last throw, resulting in 141,181,920, which is less than 162,012,510.

c) 10. Let's use a diagram to check all the possibilities:

![](https://cdn.mathpix.com/cropped/2024_05_01_09dbad5281c800d666d6g-07.jpg?height=980&width=582&top_left_y=1075&top_left_x=843)

Counting the total number of possibilities, we arrive at 10.