# ETOOBUSY ðŸš€ minimal blogging for the impatient

# A D4 from a D6, with time guarantees

**TL;DR**

Generating a D4 from a D6 with the rejection does not give guarantees on time. Can we do something about it?

In previous post a 4-faces die from a 6-faces die we saw a possible
procedure to create a *virtual D4* from a real D6, by means of the
*rejection method*.

One drawback of this method is that it cannot guarantee us how many
rolls will it take to get the next *virtual D4* roll. Sure, the
probability that it will take more than $n$ rolls is $\frac{1}{3^n}$,
which goes down fast, and yet you might have a hard limit on the
patience of your players and they might not be happy to know that 1 time
out of 27 they will need to roll *more* than three times that hated D6.

At this point, they might just start thinking youâ€™re *so cheap* ðŸ˜¡

# Letâ€™s take it from another angle

Letâ€™s forget for a moment that we have a D6 and concentrate on our goal. What are we trying to accomplish?

The answer is simple: have a device that is capable of giving us one out of four different symbols, each with probability $\frac{1}{4}$.

This would be easily accomplished with a coin, for example, because flipping it twice gives us exactly 4 different outcomes, each with our desired probability (assuming the coin is fair):

# So, letâ€™s use a coin insteadâ€¦

I guess you know where weâ€™re heading to by now. If you have a coin, you can just use that. But back to our full set of constraints, we donâ€™t have it, we have a D6 instead.

As we saw in the previous post, though, we can indeed use a D6 to
generate a *virtual* coin, and then use this twice to generate a D4.

# Putting it all together

Itâ€™s clear at this point that we can get a D4 by rolling a D6 exactly
twice. We can shortcut the *virtual coin* step and end up with the
following algorithm:

- roll a D6 once and note down the outcome as
`D6 Roll 1`

- roll a D6 once and note down the outcome as
`D6 Roll 2`

- use the table below to get the result of the D4.

D6 Roll 1 | D6 Roll 2 | D4 outcome |
---|---|---|

`1` , `2` , `3` |
`1` , `2` , `3` |
1 |

`1` , `2` , `3` |
`4` , `5` , `6` |
2 |

`4` , `5` , `6` |
`1` , `2` , `3` |
3 |

`4` , `5` , `6` |
`4` , `5` , `6` |
4 |

# Can we roll two dice once?

The table in the previous section can prompt us to think: can we roll once using two dice instead? This would save us time and we surely have another D6 somewhere in the house.

The only caveat at this point is that it MUST be possible to tell one die from the other. E.g. you might get them with different sizes, different colors, or mark one of them with a marker. If you canâ€™t do this, you will not be able to distinguish the second and the third case in the table, which will put you in trouble!

What? You donâ€™t have a marker and canâ€™t tell them apart? Wellâ€¦ roll
them and use their position with respect to the room to tell which is
the *first* and which is the *second* - in other terms, get creative!

# Better than rejection?

From an efficiency perspective, this method is worse than the rejection
method, because it *always* requires two rolls where the rejection
method requires an average of $1.5$ rolls instead.

On the other hand, this method has time guarantees, which in some cases might be more important - especially if you indeed have two (distinguishable) dice that you can roll at the same time.

So the answer is actually up to you!