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Day 9: Marble Mania
You talk to the Elves while you wait for your navigation system to initialize. To pass the time, they introduce you to their favorite marble game.
The Elves play this game by taking turns arranging the marbles in a circle according to very
particular rules. The marbles are numbered starting with 0
and increasing by 1
until every
marble has a number.
First, the marble numbered 0
is placed in the circle. At this point, while it contains only a
single marble, it is still a circle: the marble is both clockwise from itself and
counter-clockwise from itself. This marble is designated the current marble.
Then, each Elf takes a turn placing the lowest-numbered remaining marble into the circle
between the marbles that are 1
and 2
marbles clockwise of the current marble. (When the
circle is large enough, this means that there is one marble between the marble that was just
placed and the current marble.) The marble that was just placed then becomes the current
marble.
However, if the marble that is about to be placed has a number which is a multiple of 23
,
something entirely different happens. First, the current player keeps the marble they would
have placed, adding it to their score. In addition, the marble 7
marbles
counter-clockwise from the current marble is removed from the circle and also added
to the current player’s score. The marble located immediately clockwise of the marble that
was removed becomes the new current marble.
For example, suppose there are 9 players. After the marble with value 0
is placed in the
middle, each player (shown in square brackets) takes a turn. The result of each of those turns
would produce circles of marbles like this, where clockwise is to the right and the resulting
current marble is in parentheses:
[-] (0)
[1] 0 (1)
[2] 0 (2) 1
[3] 0 2 1 (3)
[4] 0 (4) 2 1 3
[5] 0 4 2 (5) 1 3
[6] 0 4 2 5 1 (6) 3
[7] 0 4 2 5 1 6 3 (7)
[8] 0 (8) 4 2 5 1 6 3 7
[9] 0 8 4 (9) 2 5 1 6 3 7
[1] 0 8 4 9 2(10) 5 1 6 3 7
[2] 0 8 4 9 2 10 5(11) 1 6 3 7
[3] 0 8 4 9 2 10 5 11 1(12) 6 3 7
[4] 0 8 4 9 2 10 5 11 1 12 6(13) 3 7
[5] 0 8 4 9 2 10 5 11 1 12 6 13 3(14) 7
[6] 0 8 4 9 2 10 5 11 1 12 6 13 3 14 7(15)
[7] 0(16) 8 4 9 2 10 5 11 1 12 6 13 3 14 7 15
[8] 0 16 8(17) 4 9 2 10 5 11 1 12 6 13 3 14 7 15
[9] 0 16 8 17 4(18) 9 2 10 5 11 1 12 6 13 3 14 7 15
[1] 0 16 8 17 4 18 9(19) 2 10 5 11 1 12 6 13 3 14 7 15
[2] 0 16 8 17 4 18 9 19 2(20)10 5 11 1 12 6 13 3 14 7 15
[3] 0 16 8 17 4 18 9 19 2 20 10(21) 5 11 1 12 6 13 3 14 7 15
[4] 0 16 8 17 4 18 9 19 2 20 10 21 5(22)11 1 12 6 13 3 14 7 15
[5] 0 16 8 17 4 18(19) 2 20 10 21 5 22 11 1 12 6 13 3 14 7 15
[6] 0 16 8 17 4 18 19 2(24)20 10 21 5 22 11 1 12 6 13 3 14 7 15
[7] 0 16 8 17 4 18 19 2 24 20(25)10 21 5 22 11 1 12 6 13 3 14 7 15
The goal is to be the player with the highest score after the last marble is used up.
Assuming the example above ends after the marble numbered 25
, the winning score is 23+9=32
(because player 5 kept marble 23
and removed marble 9
, while no other player got any points
in this very short example game).
Here are a few more examples:
10
players; last marble is worth1618
points: high score is8317
13
players; last marble is worth7999
points: high score is146373
17
players; last marble is worth1104
points: high score is2764
21
players; last marble is worth6111
points: high score is54718
30
players; last marble is worth5807
points: high score is37305
What is the winning Elf’s score?
Part Two
Amused by the speed of your answer, the Elves are curious:
What would the new winning Elf’s score be if the number of the last marble were 100 times larger?