# Sophawa's Formula

Sophawa 庙 (U • C • T) formula is a formula to calculate the time remaining until a hero's temple is completed.

## Formula and derivation

### Formula and use

To calculate the time remaining to complete a temple, use this formula:

**R = [ (100 * A ) / T ] - A**

Where:

**R**is the time remaining to complete the temple.**A**is the age of the hero. If the age is in days, the result will be in days (and so on).**T**is the percentage of the temple that is complete (in numerical form). For example, 78.3% would be 78.3.

### Derivation

Let's assume the brick-gathering rate is constant. We can deduce the percentage of bricks collected is proportional to the time passed since the birth of the hero, from which we can write :

**100/B=T/A**

It means that you collect the bricks for the whole temple (100 bricks) at the same rate (divided by the time from the beginning to the completion, named B) as you collected bricks since the beginning. So :

**1/B=T/100A**

**B=100A/T**

However, you already started to collect bricks, so you have to substract the time you already passed :

**R=(100A/T)-A**

## Justification and limits

### Spode's derivation

The god Spode wrote a formula (well-named the Spode's Formula) to help gods who made many arena fights regularly to have a more accurate value. However, his original derivation was wrong, and the formula revealed unefficient. Here is a corrected derivation, that shows the power of Sophawa's formula :

First of all, let us calculate the rate at which a hero gains bricks from quests and other random events, not including bricks gained from arena duel wins.

[T-(W/10)]/A

Where A is the age of the hero, T is the percentage of temple completion and W is the number of wins. The number of wins is divided by ten since one win will reward one gold brick and one gold brick equals 0.1 percent of temple completion. We must have conservation of units. The answer from this formula will therefore be measured in temple completion percentage per unit of time used.

Now we need to calculate the rate at which a hero duels. This is the age over the total number of duels as given by this formula:

(W+L)/A

Where L is the number of duel losses. The answer from this formula is measured in duels per unit of time used. However, only a certain amount of these duels per unit of time will reward with a gold brick. This amount can be gained from the fraction of wins over total duels. Therefore, the players history is taken into account. This is the fraction:

W/(W+L)

Now we must multiply the rate of duels with this fraction to give the final rate of duel wins.

[W/(W+L)]*[(W+L)/A]

Simplified:

W/A

Let's now divide by ten to get the rate of bricks gain from duels:

W/10A

Now to get the total brick gain rate we add the two main rates together:

[[T-(W/10)]/A]+(W/10A)

Simplified:

T/A

This rate is measured in temple completion percentage per unit of time so if we want to know how long it will take the hero to complete the temple we must use this formula:

100/(T/A)=100A/T

However the hero has already taken some time and thus completed some temple so we must subtract the hero's age from this to reach the final formula:

(100A/T)-A

Thus, this derivation shows that the Sophawa's formula factor in, at the same time, the random events (because of the law of large numbers) and the influences of the god at the condition these influences are made at a roughly regular rate.

### Flaws of the formula

This formula can estimate a roughly accurate time to go, although it has these problems:

- It makes the assumption that the brick-gathering rate is consistent, which can be a problem for gods who do not have a regular presence, or who change their way of playing.
- It is as precise at the unit of time you use

As these are impossible to overcome without lots of data-gathering, it is best to take the result with a pinch of salt. The formula is difficult to error, and as of yet, no-one has spotted any problems. The formula implies that brick gain by different means do not affect the overall brick-rate, such as a 3-month-old hero who gained 50% of 100 bricks from the Arena does not have a different brick-rate than a hero who gained 100% of bricks from the Arena or 0% from the Arena, provided that the amount of bricks and the age are the same and that the two continue to duel and be influenced at the same rate as before. Also, complicated formulas with multiple factors are likely to be ridiculously difficult, and to use probabilitities that the Great Random hasn't revealed us.

## Aerian's derivation

The problem of the constant-assumed brick gain rate can be corrected by a simple experiment:

1)At a precise date, note your completion percentage

2)Wait some days, trying to battle, influence the hero, etc. at the same rate as you will after the experiment

3)After a fixed number of days (the more you take, the more precise the formula will be), note the new percentage of temple completion, and use this formula :

**T=(100-P)*dA/dP**

with dA the difference between the time of the start and the time of the end, and dP the difernce between the percentage of completion between the start and the end.

The interest is that, if you change radically your way of playing (for example, if you come back after a long time), the gain rate will be fixed on your new rate, and will not count the past events.

However, it is necessary to note that, even if it can be more precise that the original formula, it still need to be taken with perspective, as the random events occur at a rate that can invalid the use of the law of large numbers. Thus, it can produce a result that might be far from the reality, even if it must be more correct than the one of the original formula.