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Inverted Dice is a challenging tactic dice game for smart adults and curious children. The game introduces a new way to look at dice rolls – a way to regard five normal six-sided dice as one big twenty-sided die (with some odd probability properties). Normally, you would throw some dice and use the sum of the numbers being shown. In this game, you use the sum of the numbers not being shown:

Such a sum is called an inverted dice sum.
You need five dice to play. The structure of the game is a bit like the one in Yahtzee, with a similar score chart and three rolls (or fewer) per turn.
There are twenty rounds in the game. In each round, players take turns rolling five dice, trying to reach one of the inverted sums 1 to 20 (it has to be a number that the player has not previously achieved or zeroed out).
After each inverted dice roll, the player chooses which dice to keep, and which to reroll. A player may reroll some or all of the dice up to two times on a turn, making a maximum of three rolls each turn.
If a player achieves a

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