:Draft:Inverted Dice

An inverted dice sum of a roll is the sum of the values that are not present in the roll. To find the inverted sum, simply add together the values that you do not see (first method in the examples below).

However, there are two different ways to find the inverted sum of a roll, since it can also be calculated as 21 minus all unique values that are seen in the roll (second method in the examples below).

• Dice value 2Dice value 3Dice value 5Dice value 5Dice value 6 is 5 since Dice value 1+Dice value 4 is 5 (neither Dice value 1 nor Dice value 4 is shown).

• Dice value 1Dice value 1Dice value 4Dice value 5Dice value 6 is also 5 since Dice value 2+Dice value 3 is 5.

• Dice value 4Dice value 4Dice value 5Dice value 5Dice value 6 is 6 since Dice value 1+Dice value 2+Dice value 3 is 6.

• Dice value 1Dice value 1Dice value 2Dice value 2Dice value 3 is 15 since Dice value 4+Dice value 5+Dice value 6 is 15.

• Dice value 1Dice value 2Dice value 2Dice value 3Dice value 3 is 15 since Dice value 1+Dice value 2+Dice value 3 is 6 (only count each shown value once), and 21 – 6 is 15.

• Dice value 1Dice value 1Dice value 1Dice value 2Dice value 2 is 18 since Dice value 1+Dice value 2 is 3, and 21 – 3 is 18.

• Dice value 1Dice value 2Dice value 2Dice value 2Dice value 2 is also 18 since Dice value 1+Dice value 2 is 3, and 21 – 3 is 18.

• Dice value 5Dice value 5Dice value 6Dice value 6Dice value 6 is 10 since Dice value 5+Dice value 6 is 11, and 21 – 11 is 10.

In a five-dice roll, the lowest inverted sum is 1 (Dice value 2Dice value 3Dice value 4Dice value 5Dice value 6) and the highest is 20 (Dice value 1Dice value 1Dice value 1Dice value 1Dice value 1).

There are twenty rounds in the game. In each round, players take turns rolling five dice, trying to achieve the scores in the table, i.e., one of the inverted dice sums 1 to 20. A result can be scored if it has not previously been scored (or crossed out) by the player.

After each 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 per turn.

If a player achieves an available result, it is registered in the score chart. If the player fails (after three rolls), a number must be recorded as zero points. Thus, every player must put either a score or a zero into a score box each turn. Each result can only be scored once in every game.

The game ends when all score boxes are used. The player with the highest total score wins the game.

Each inverted dice sum that a player scores is worth the same amount of points as the inverted sum, e.g., both Dice value 6Dice value 6Dice value 6Dice value 6Dice value 6 (inverted sum 15) and Dice value 1Dice value 1Dice value 1Dice value 2Dice value 3 (also 15) can be scored as 15 points.

Players can receive up to three bonuses by achieving all scores in one or more of the following bonus sections:

• Low bonus section: The scores 1 to 5. Getting these five results gives 50 points in the upper bonus row.
• Middle bonus section: The scores 6 to 15. Getting these ten results gives 50 points in the middle bonus row.
• High bonus section: The scores 16 to 20. Getting these five results gives 50 points in the bottom bonus row.

The highest possible score is 360, including all three bonuses.

• Board Game Arena is a free multiplayer platform where Inverted Dice can be played online with up to 5 players.{{Cite web|title=Play Inverted Dice (BGA)|url=https://boardgamearena.com/gamepanel?game=inverteddice|website=boardgamearena.com}}
• The publisher has a website where Inverted Dice can be played online with up to 3 players.{{Cite web|title=Inverted Dice™ – Play online|url=https://www.simonjensen.com/InvertedDice/?players=1&help=3|website=simonjensen.com}}

Inverted dice sums can be treated mathematically.{{Cite web|title= Jensen, S. K. (2023). A short introduction to the theory of inverted dice sums.|url=https://www.researchgate.net/publication/380030580|website=ResearchGate}} This involves problems concerning partitions and multisets.

Following are the probabilities of scoring inverted dice sums in the initial throw of five dice.

The inverted sum with the highest probability is 9. It is worth noticing, that there are five pairs of inverted sums with equal probability. These probability pairs are P(1)=P(2), P(3)=P(4), P(10)=P(11), P(17)=P(18), and P(19)=P(20).

Since both 19 and 20 are equally hard to get, the first value to be crossed out is often 19, followed by 20.

{{reflist}}

• [https://www.simonjensen.com/InvertedDice/ Official website] - Background, score sheets, online game
• [https://www.simonjensen.com/pdf/Rules_Inverted_Dice.pdf Rules of Inverted Dice] - Rules, examples
• {{bgg title|190950|Inverted Dice}} - Description, forums, stats, ratings
• [https://boardgameguys.com/inverted-dice/ Inverted Dice (2015) at Board Game Guys] - Review
• [https://boardgamearena.com/gamepanel?game=inverteddice Inverted Dice at BGA] - Online game, rankings

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