User blog:BunsenH/How many days to Celestial? A math "trick"

People have sometimes wondered about how long it takes to complete a Celestials' constellation in the DoF Daily Login game. On average, of course.

The answer can take us to a very useful problem-solving method: Dimensional analysis. I learned about it in my last year of high school, I think. In various teaching jobs I've held, I have often found that even many upper-year university students have never learned about it. But it is actually pretty straightforward. It works by a series of multiplications, converting one set of units to another.

Let's start with a few things we already know. Each constellation has 24 steps to complete. Each roll moves the Celestial by 11⁄9 steps, on average. And each day, you get one free roll, with an extra roll every 5 days.

That "11⁄9 steps per roll" means that on average, 9 rolls give you 10 steps. One way of looking at it is that 10 steps are equivalent to 9 rolls.

10 steps &equiv; 9 rolls

We can divide both sides of the equation by 10 steps. It is very important to keep the units through all of these calculations! They are an important clue to the steps that must be followed, as well as helping you catch any mistakes. Doing that, we get

$$1 \equiv{{\rm 9\ rolls}\over{\rm 10\ steps}}$$