The game Inverted Dice™

My initial thought was that I couldn't possibly be the first to have thought of this, so 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 on 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 think the game is 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 sheet 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 please, get in touch if you're interested).

Do you want to spread the game?  Download the rulesDownload the rules from www.simonjensen.com/InvertedDice.

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

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

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

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

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 sums

Normally, you roll some dice and calculate the sum of the values being shown. In this game, you calculate the sum of the values not  being shown.

Such a sum is called an inverted dice sum, and the corresponding roll is called an inverted dice roll. The rolls are made with standard dice (six-sided).

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.

You need five dice to play, which means there are twenty possible results (1 to 20), most of which can be achieved in many different 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 once). Then, subtract this sum from 21.

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.

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. The rules can be downloaded here.

There are twenty rounds in the game. In each round, players take turns rolling five dice, trying to reach one of the inverted dice sums 1 to 20 (it has to be a number that the player has not previously achieved or zeroed out).

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 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.

Each player must record either a score or a zero in a score box every turn.

The game ends when all score boxes have been filled. The player with the highest total score wins the game.

Bonuses

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

• Low bonus section is the scores 1 to 5. These five numbers will earn you 50 points in the first bonus row.
• Middle bonus section is the scores 6 to 15. These ten numbers will earn you 50 points in the second bonus row.
• High bonus section is the scores 16 to 20. These five numbers will earn you 50 points in the third bonus row.

Masters

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. A player who achieves this is awarded with the rare, lifelong and prestigious title Inverted Grandmaster.

Hints

It is rather easy to get the middle bonus. The low 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 these large numbers at the expense of the low 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.

Inverted Dice™ is perfect 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 particularly 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. New players often need to play it at least fifty times before they begin thinking about strategy.

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