Draft:Inverted Dice

Not to be confused with the mathematical concept of Intransitive dice.

Inverted Dice

Inverted Dice, box and score sheet
Designers	Simon K. Jensen
Publishers	OffCircle
Publication	2016
Players	1–5
Playing time	10–30 minutes
Age range	12+
Skills	probability strategy luck

Inverted Dice is a dice game invented by Simon K. Jensen and released by OffCircle in 2016. The game introduces the concept of inverted dice sums, i.e., sums of the dice values that are not present in a roll.

The structure of the game resembles that of Yahtzee (and variants such as Yatzy), with five dice, and three rolls (or fewer) per turn.

Inverted dice sums

An inverted dice sum of a roll is the sum of the values that are not seen in the roll.

However, there are two different ways to find the inverted sum of a roll, since it can also be calculated as 21 minus each unique value that are seen in the roll.

Examples

First method: Summing up values that are not shown:

• is 5 since + is 5 (neither nor is shown).

• is also 5 since + is 5.

• is 6 since ++ is 6.

• is 15 since 4 + 5 + 6 is 15.

Second method: 21 minus each unique shown value:

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

• is 18 since + is 3, and 21 – 3 is 18.

• is also 18 since + is 3, and 21 – 3 is 18.

• is 10 since + is 11, and 21 – 11 is 10.

In a five-dice roll, the lowest inverted sum is 1 () and the highest is 20 ().

Gameplay

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.

Scoring

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

Bonuses

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.

Maximum score

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

Digital versions

• Board Game Arena is a free multiplayer platform where Inverted Dice can be played online with up to 5 players.
• The publisher has a website where Inverted Dice can be played online with up to 3 players.

Mathematics

Inverted dice sums can be treated mathematically. This involves problems concerning partitions and multisets.

Probabilities

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

Inverted dice sum	Probability	Probability in %
1	120/7776	1.543%
2	120/7776	1.543%
3	360/7776	4.630%
4	360/7776	4.630%
5	600/7776	7.716%
6	750/7776	9.645%
7	870/7776	11.188%
8	780/7776	10.031%
9	930/7776	11.960%
10	720/7776	9.259%
11	720/7776	9.259%
12	510/7776	6.559%
13	360/7776	4.630%
14	240/7776	3.086%
15	211/7776	2.713%
16	61/7776	0.784%
17	31/7776	0.399%
18	31/7776	0.399%
19	1/7776	0.013%
20	1/7776	0.013%

The inverted sum with the highest probability is 9.

External links

• Official website - Background, score sheets, online game
• Rules of Inverted Dice - Rules, examples
• Inverted Dice at BoardGameGeek - Description, forums, stats, ratings
• Inverted Dice at BGA - Online game, rankings

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Categories:

• Sequence dice games