Introduction
Inverted Dice® is a challenging tactic dice game for smart people. 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 probabilities).
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.
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. There are twenty possible results (1 to 20), some of which can be achieved in several ways.
How to play the game
The structure of Inverted Dice® is similar to games like Yahtzee, Yatzy and Yacht, with a 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 from 1 to 20 (it must 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 turn ends when the player puts either a score or a zero into a score box.
The game ends when all score boxes are used. The player with the highest total score wins the game.
Bonuses
Players can receive bonuses by achieving all scores in one or more of the following bonus sections.
Top bonus is 1 to 5. Getting all of these five numbers will earn you 50 points in the upper bonus row.
Middle bonus is 6 to 15. Getting these ten numbers will earn you 50 points in the middle bonus row.
Bottom bonus is 16 to 20. Getting 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, and the player is awarded with the lifelong and prestigious title Inverted Grandmaster.
How to calculate
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 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 in the roll.
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 5  is 9 because 1 2 6 are not shown, and because 1 + 2 + 6 = 9.
1 1 4 4 4  is 16 because 2 3 5 6 are not shown, and the sum of these numbers is 16.
1 1 1 4 4  is also 16 because again 2 3 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 since 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.
2 3 4 5 5  is 7, since  2 + 3 + 4 + 5 = 14  (count each value once), and then 21 14 = 7.
1 3 4 5 5  is 8, since  1 + 3 + 4 + 5 = 13, and then 21 13 = 8.
3 3 4 5 5  is 9, since  3 + 4 + 5 = 12, and then 21 12 = 9.
1 2 4 5 5  is 9, since  1 + 2 + 4 + 5 = 12, and then 21 12 = 9.
2 2 4 6 6  is 9, since  2 + 4 + 6 = 12, and then 21 12 = 9.
3 3 5 5 5  is 13, since  3 + 5 = 8, and then 21 8 = 13.
1 1 2 3 3  is 15, since  1 + 2 + 3 = 6, and then 21 6 = 15.
1 1 5 5 5  is 15, since  1 + 5 = 6, and then 21 6 = 15.
2 2 2 4 4  is 15, since  2 + 4 = 6, and then 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.
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 rewarding to go for these large numbers instead 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.
The easiest way to achieve 18  is to get nothing but ones and twos.
The easiest way to achieve 17  is to get nothing but ones and threes.
The easiest way to achieve 16  is to go for either twos and threes  or  ones and fours.