Addition (April 2014):
The game Inverted Dice™ has been released online, please visit www.inverteddice.com.
Rules for Inverted Dice™ are available HERE.
Included in the rules are some examples of inverted dice which are not included in the text below.
This text concerns standard 6-sided dice. One might think of various extensions of the concept that follows and then deal with other types of dice. But for the moment, we restrict ourselves to normal dice with 6 faces, numbered from 1 to 6.
When rolling two dice, it is common (in hundreds of different games) to look at the sum of the numbers shown on the upper surfaces. As you know, there are 11 possible results (2 to 12) from a roll with 2 dice. My idea is to look at the sum of the numbers not shown on any dice. In that way we get 11 different possible results (namely the numbers from 10 to 20, i.e. with two dice).
I have chosen the name inverted dice for this way of interpreting a roll of dice.
Some examples are required.
Playing with 1 inverted die
This case is the least interesting, but it can without doubt be useful:
When playing with 1 inverted die, you can achieve any result from 15 to 20 (that is 6 possible results).
You have a chance of 1/6 for each of these results, just as you have in a roll the traditional way.
This can be used in several new (not yet invented) types of games, or in modifications of existing games.
For instance, you might imagine a game in which you move your token after rolling a die, and sometimes (in specific situations) you get to move after an inverted die roll (which is a significantly larger number).
Maybe even games like Ludo might be made stimulating in the future.
Playing with 2 inverted dice
Now it gets much more interesting. That is because we find a radical change of the probabilities.
But first, let’s see some examples:
When playing with 2 inverted dice, you can achieve any result from 10 to 20 (that is 11 possible results).
We realise that there is only one way of getting the result 20, and that is when both the dice show the number 1. However, we also see that for instance the result 18 can be achieved in several different ways.
It is time to make a complete table to get an overview. We have 36 possible outcomes to look at:
We can now without difficulty create two tables showing the odds of all possible results, in the case of both two normal dice and two inverted dice:
One interesting characteristic of two inverted dice are that the two highest numbers are also those most difficult to achieve. This is usually not the case, since it’s normally just as hard to roll 2 as 12 with two dice.
I can think of several uses for this. For example, try playing Monopoly with inverted dice instead of regular dice (or change the rules and add the possibility of a player having to use inverted dice in certain situations). Feel free to invent new games of your own.
Playing with 3 inverted dice
Here it gets more interesting. This is caused by the fact that with 3 inverted dice you don’t have the same number of possible results as with 3 normal dice.
With 3 normal dice, the sum is always between 3 and 18 (i.e. 16 possible results), but with 3 inverted dice, the sum is always between 6 and 20 (i.e. only 15 possible results).
That this is the case, we can easily deduct, since we get the lowest result when only the three largest dice values, being 4, 5 and 6, are shown. The sum is then 1+2+3 = 6. In the same way we see, that we get the largest result when only the number 1 is shown, i.e. 2+3+4+5+6 = 20.
We could create a table showing all possibilities of achieving the results between 6 and 20 using three inverted dice, but to do so would be a rather tedious task.
As an example, I can demonstrate that the result 11 can be achieved with the following combinations:
I’ll leave it to my readers to calculate all the odds for dice rolls with 3 inverted dice. This can be done manually with the help a table similar to Table 1, but this is quite a lot of work (some programming skills would be useful). There are 6 times 6 times 6 possible outcomes, so the table consists of 216 rows.
Playing with 4 inverted dice
Normally you will have 21 different possible results (from 4 to 24) from a roll using 4 dice, but with an inverted roll we will only have 18 possible results (from 3 to 20). I shall leave to my reader to look at this case (and come up with new games).
Playing with 5 inverted dice
This case is extremely interesting. Firstly, there are exactly 20 possible results from a roll using 5 inverted dice, and the results are between 1 and 20. In other words:
Five 6-sided dice can together form one 20-sided die
(with a very peculiar probability distribution).
Secondly, 5 is also the exact number of dice used in a regular game of Yahtzee. I have attempted to develop a kind Inverted Yahtzee, but since Yahtzee is not based on sums, but rather patterns and numbers, it is hard to invert this game.
Following are a few examples of rolls using 5 inverted dice (keep in mind that the result is the sum of the numbers not being shown):
When calculating the result of an inverted dice roll, it can be useful knowing that the sum of the numbers from 1 to 6 is 21. For instance, if you only rolled threes and fives, you know that the inverted sum is 13, since 3 plus 5 equals 8, and 21 minus 8 equals 13. It is sometimes easier to count that way.
Playing with 6 inverted dice
This case is also interesting. It is the first case which has 21 different possible results and where it’s also possible to achieve the result zero. I will leave all details for the reader. Perhaps some student of mathematics will like to delve further into this.
Playing with many inverted dice
Before finishing this text, I would like to present a small table showing the number of possible results of rolls with n dice, both regular and inverted.