Rules of Hog

In Hog, two players alternate turns trying to be the first to end a turn with at least 100 total points. On each turn, the current player chooses some number of dice to roll, up to 10. That player's score for the turn is the sum of the dice outcomes.

However, a player who rolls too many dice risks:

  • Pig Out. If any of the dice outcomes is a 1, the current player's score for the turn is 1.
    • Example 1: The current player rolls 7 dice, 5 of which are 1's. The player scores 1 point for the turn.
    • Example 2: The current player rolls 4 dice, all of which are 3's. Since Pig Out did not occur, they score 12 points for the turn.

In a normal game of Hog, those are all the rules. To spice up the game, we'll include some special rules:

  • Picky Piggy. A player who chooses to roll zero dice scores the \( n \)-th digit of the decimal expansion of \( 1/21 (0.04761904\dots) \) where \( n \) is the opponent's score. As a special case, if \( n \) is 0, the player scores only 1 point.
    • Example 1: The current player rolls zero dice and the opponent has a score of 3. The third digit of the decimal expansion of \( 1/21 \) is \( 7 \), the current player will receive 7 points.
    • Example 2: The current player rolls zero dice and the opponent has a score of 13. The 13th digit of the decimal expansion of \( 1/21 \) is \( 0 \), the current player will receive 0 points.
    • Example 3: The current player rolls zero dice and the opponent has a score of 0. The current player will receive 1 point.
  • Feral Hogs. A player gets 1 extra point for each dice whose outcome has the same parity (i.e. both odd or both even) of the player's score.
    • Example 1: The current player has a score of 21 and rolls three dices. The outcomes are 1, 3, 5. The player will get \( 1 + 3 = 4 \) points due to the Pig Out rule.
    • Example 2: The current player has a score of 40 and rolls four dices. The outcomes are 2, 3, 4, 5. The player will get \( 2 + 3 + 4 + 5 + 2 = 16 \) points.
    • Example 3: The current player has a score of 60 and rolls one dice. The outcome is 3 and the player will get 3 points.
  • Swine Swap. Define the excitement of the game to be three to the power of the sum of both players' scores. After points of the turn are added to the current player's score, if the excitement's first digit and last digit are the same, the scores of both players should be swapped.
    • Example 1: At the end of a turn, the players have scores of 2 and 4. Since \( 3^{2 + 4} = 729 \), and \( 7 \neq 9 \), the scores are not swapped.
    • Example 2: At the end of a turn, the players have scores of 11 and 1. Since \( 3^{11 + 1} = 531441 \), and \( 5 \neq 1 \), the scores are not swapped.
    • Example 3: At the end of a turn, the players have scores of 23 and 4. Since \( 3^{23 + 4} = 7625597484987 \), and \( 7 = 7 \), the scores are swapped.