# The game Inverted Dice™

For more than 5000 years, humankind has been using dice — in play and war, in decision-making and ancient trials, as pastime activity, as mathematical aids. Dice have been included in great literary works, attributed divine powers, magic, wisdom. We have sung about dice, worshiped dice. And above all, *played* with dice.

3 4 6 6 6 is 8

1 2 3 6 6 is 9

1 6 6 6 6 is 14

Can you see why?

These little random generators have followed us throughout our long history. Thus, I was surprised when I found a *new* way to interpret dice rolls, back in 2012. My initial thought was that I couldn't possibly be the first to have thought of this.

## + More background story

I continued investigating various aspects of the concept of *dice*. Over a couple of months, I read everything I could find about both dice games and the mathematics behind them, but nowhere did anything resembling *inverted dice sums* come up. It turned out to be indeed a new idea.

All of this resulted in a game for which I received public funding. In 2016, *Inverted Dice™* was released, both online and in the form of a game box in a limited first volume of 500 copies.

My former company OffCircle ran this project on the sideline until the end in 2021. At that time, there were around a thousand registered players on the website, and only a few hundred game boxes had been distributed. I once read that when *Yahtzee* was launched at a large scale in the 1950s, it took several years (and a large marketing budget) to make it self-spreading, and decades to establish it as a classic, timeless dice game.

So, as a conclusion (not necessarily final), I don’t consider the game to be a fiasco, even though it might not be more than 1% of the world’s population who would potentially enjoy it (the most nerdy 1%). If properly distributed (maybe together with *Yahtzee*), it has potential. Every year, at least 50 million copies of *Yahtzee* are sold worldwide. Why not print an *Inverted Dice* score card on the backside of each *Yahtzee* sheet and sell the kit for a few extra cents? Whatever the case, I don’t have the kind of connections needed to implement this idea (but get in touch if you're interested).

### Before you download

This work is protected by Swedish Law (1960:729) and international laws concerning copyrights of literary and artistic works. Editing the document (including extending and/or reducing the content) is prohibited. The document may be used freely for private reading. The document may also be printed in connection with education, and the content may be used as instructional material provided that the author (Simon K. Jensen) is credited and applicable rules regarding compensation for the use of copyrighted works are applied. Other forms of replication and/or distribution are *strictly prohibited*.

Do you own a copy of the game? Download extra score sheets from www.simonjensen.com/InvertedDice.

Do you want a copy of the game? Send a message to information@simonjensen.com.

Do you want to know more about the mathematics behind the game? Download a paper from www.simonjensen.com/texts#mathematics.

Below you will find an introduction to the concept, the rules of the game, and an online version with tutorial features for beginners.

## + About inverted dice rolls

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).

This new way of interpreting dice rolls is called *inverted dice*, and the principle can easily be applied to all kinds of dice (not only *six-sided*) and all kinds of dice rolls (not only rolls with *five* dice).

###### Inverted dice rolls

Normally, you would throw some dice and calculate the sum of the numbers being shown. In this game, you calculate the sum of the numbers ** not** being shown.

Such a roll is called an *inverted* dice roll. In the game Inverted Dice™ all inverted dice rolls are made with standard six-sided dice. You need *five* dice to play.

To make an inverted dice roll, just roll your dice. The result is the sum of the values that are not included in your roll. With five dice, you have twenty possible results (1 to 20), most of which can be achieved in several ways.

###### Calculating inverted dice sums

When calculating the result of an inverted dice roll, always remember these two facts:

- The result is the sum of the dice values that are
*not*included in the roll. - The sum of all numbers from 1 to 6 is 21.

This gives us *two* smart ways of calculating an inverted sum.

The first method (Sum of values that you do * not* see) is straightforward. Simply add the dice values that are not present amongst the five dice.

The second method (**21** minus values that you * do* see) is often a quicker way. First, calculate the sum of all values that

*are*included in the roll (but only count each value

*). Then, subtract this sum from 21.*

**once**###### Examples (the first method)

The result of an inverted dice roll is the sum of the numbers ** not** being shown.

1 2 3 4 5 is 6 because 6 is the only number *not* being shown.

1 2 4 5 6 is 3 because 3 is the only number *not* being shown.

3 3 4 5 6 is also 3 because 1 and 2 are *not* shown, and 1 + 2 = 3.

3 3 4 5 6 is 9 because 1 2 6 are *not* shown, and 1 + 2 + 6 = 9.

1 1 2 3 3 is 15 because 4 5 6 are *not* shown, and 4 + 5 + 6 = 15.

1 1 1 2 3 is also 15 because again 4 5 6 are *not* shown.

2 3 4 5 6 is 1 because 1 is the only number *not* being shown.

1 1 1 1 1 is 20 because 2 + 3 + 4 + 5 + 6 = 20.

###### Examples (the second method)

The result is 21 *minus* the numbers being shown. Only count each shown value ** once**.

4 5 5 6 6 is 6 since 4 + 5 + 6 = 15 (only count each value *once*), and 21 − 15 = 6.

1 3 4 5 5 is 8 since the sum of 1 3 4 5 is 13, and 21 − 13 = 8.

3 3 4 5 5 is 9 since 3 + 4 + 5 is 12, and 21 − 12 = 9.

1 2 4 4 5 is 9 since 1 + 2 + 4 + 5 is also 12, and 21 − 12 = 9.

2 2 4 6 6 is 9 since 2 + 4 + 6 = 12, and 21 − 12 = 9.

3 3 5 5 5 is 13 since 3 + 5 = 8, and 21 − 8 = 13.

1 1 2 3 3 is 15 since 1 + 2 + 3 = 6, and 21 − 6 = 15.

1 1 5 5 5 is 15 since 1 + 5 = 6, and 21 − 6 = 15.

2 2 2 4 4 is 15 since 2 + 4 = 6, and 21 − 6 = 15.

6 6 6 6 6 is 15 since 21 − 6 = 15.

5 5 5 5 5 is 16 since 21 − 5 = 16.

1 1 3 3 3 is 17 since 21 − 4 = 17.

1 1 2 2 2 is 18 since 21 − 3 = 18.

2 2 2 2 2 is 19 since 21 − 2 = 19.

## + How to play the game

The *structure* of the game is very similar to that of Yahtzee, with a similar score chart, five dice, 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 numbers (inverted) from 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 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.

Every player must put *either* a score *or* a zero into a score box each turn.

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

###### Bonuses

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

- Top bonus section is the scores 1 to 5. These five numbers will earn you
*50 points*in the*upper*bonus row. - Middle bonus section is the scores 6 to 15. These ten numbers will earn you
*50 points*in the*middle*bonus row. - Bottom bonus section is the scores 16 to 20. These five numbers will earn you
*50 points*in the*bottom*bonus row.

###### Completing the circle

A player who reaches 290 points earns the right to be referred to as *Inverted Master*.

The highest possible score a player can achieve is 360. This is referred to as having *completed the circle* (a circle is 360°), and the player is awarded with the lifelong and prestigious title *Inverted Grandmaster.*

###### Hints

It is rather easy to get the middle bonus. The top bonus is also within reach, but so are the scores 16, 17 and 18 (that’s 51 points right there) so it might be a good strategy to go for the large numbers at the expense of the top bonus.

When counting your final score, remember that the total sum of all numbers from 1 to 20 is 210. You only have to subtract the scores that were *zeroed out* from 210. After that, you add the bonuses (either 0 points, 50 points, 100 points or 150 points).

To achieve 20, you need to roll 1 1 1 1 1 which is hard to get.

To achieve 19, you need to roll 2 2 2 2 2 (same probability as 20).

The easiest way to achieve 18 is to get only *ones* and *twos*.

The easiest way to achieve 17 is to get only *ones* and *threes*.

## + Play the game online

Inverted Dice™ is a game for cozy evenings with the family or some good friends. It's a game for people who do not want to stare at a screen (at least not while playing dice).

It is a game where neither 6 6 6 6 6 nor 5 5 5 5 5 are particularily good rolls. When played in cafés or other public places, the game usually confuses any spectator who happens to pass by, because it is impossible to figure out the rules solely by observation. Yet, the rules are simple to explain.

So, it's a *cool* dice game that should be played with *real* dice.

But to master the game requires extensive practice. It is common to have to play it at least fifty times before you can even *start* thinking about strategy.

A well-trained player *always* gets at least 200 points, while a beginner often lands below 100 points.

When practising, you can use the online version below.

###### Choose options

? |
Click this symbol to show an explanation of the latest roll, or the possible outcomes for re-rolls with a single die. |

i |
This symbol marks results, or choices to make before the game can continue. |

T |
Orange text shows the players' name, turn, and total score. |

###### Choose number of players

Inverted Dice™ | Score | ||

One (1) |
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Two (2) |
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Three (3) |
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Four (4) |
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Five (5) |
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BONUS (50 points) |
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Six (6) |
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Seven (7) |
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Eight (8) |
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Nine (9) |
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Ten (10) |
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Eleven (11) |
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Twelve (12) |
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Thirteen (13) |
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Fourteen (14) |
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Fifteen (15) |
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BONUS (50 points) |
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Sixteen (16) |
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Seventeen (17) |
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Eighteen (18) |
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Nineteen (19) |
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Twenty (20) |
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BONUS (50 points) |
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TOTAL |