# Pet Levels

Pets try to copy their owners in every way possible, and since they live and grow older as they gain more experience, they also have levels just like heroes. However, whilst heroes increase their levels by gaining more experience, pets increase their levels entirely by growing older. The growth rates of pets are not currently well understood but it is all under investigation.

This is the graph you get if you plot levels against age:

The trend-line passes through the top points for each level (the maximum age to have the level before it increases) and the curve is followed on despite the miscorrelation with higher level points. This is because as age increases, it becomes more approximate as it says 'about *** months' or something similar, so these points can just be used as a rough guideline. The main point is that it conclusively proves that pet levels are decided entirely by age. Now we just need the formula. It seems to be a power curve of some sort.

What do the vertical strips of dots mean then? each 'strip' is the band of ages which a pet must fit into to be that certain level. If pets of ALL ages had been plotted on the graph, the vertical strips would reach the trend line, but most do not since there are missing gaps in the data. However, the fact that some dots (each representing individual pets) are on the same level (ie. have the same age) but are in different strips (ie. have different levels) implies that the gradient of the growth curve varies with different species of pets (thus they grow at different rates). These anomalies usually occur near the top and bottom of the strips as would be expected since it is impossible to distinguish between the actual age bands and other growth curves (different age bands) superimposed on them, although they both seem to exist.

• Hypothesis 1: From the stats that people have shared, we can see that the gradient of the pet growth curve is directly proportional to the level a hero must be to tame the specific pet species. For example, a Biowolf will level up faster than a Dust bunny.

In order to test this hypothesis we would have to plot the growth curves of many species of pets, which is not a task many would wish to undertake since it would be a lengthy process. However, we do in fact have the full growth data for one particular pet at the time of writing; the Solar bear. The graph below shows this data as originally plotted.

A rough line of best fit has been drawn over the top but we can clearly see that there is an underlying pattern of curves which has been ignored. Someone else decided that since a pet is just an animal, the pet growth curves would be a bit more like a real growth curve for an animal (a bit more curvy), so they decided to look for a better underlying trend by cumulating the days and putting this age of the pet/time since pet birth on the x-axis and the level on the y-axis, like might otherwise usually be expected. The result is shown below.

We can see that there is a definite power trendline with the equation L = 1.98t0.521 where L is the level of the pet and t is its age (in days). You may instantly be concerned with the constants involved since you might expect them to be 'nicer' numbers as in other algorithms employed in the game of Godville. Our first hypothesis states that if were to plot the growth curve of another species, we would maybe get the same coefficient but a different exponent (to produce a different rate of gradient change), so we will know how accurate it is once somebody has done this.

• Hypothesis 2: The growth curve of any pet follows a power trendline of the form L = Gtk where G is a growth coefficient and k is a growth exponent (L and t already defined). The growth exponent varies with pet species while the coefficient stays the same. It is not known how exactly the exponent may vary, if at all.

To test Hypothesis 1 & 2 we must plot the equivalent growth curve for a pet species available to heroes at a lower [hero] level. We would then see if indeed different pet species have different growth curves (Hypothesis 1) by looking at the resulting growth exponent and coefficient, if the growth curves of all pet species do in fact follow a power trendline (Hypothesis 2) since this has yet to be fully established (the maths may look right but there could be a more fundamental formula which remains undiscovered as yet).

Well, somebody finally appears to have collected lots of data for many pet species. All of this data can be found on the talk page of this article The pet species in this assortment of information for which we have by far the most data is the Double Dragon so the time was cumulated and the same kind of growth curve plotted as before to give the following graph.

We can immediately see a power curve again, providing strong evidence for hypothesis 2, but we see an anomalous curvy segement at the start so what is this? If you look at the original data, not shown on the graph are the periods of time in which the pet was dead. The first point of death is at day 35 and there is a noticable corresponding kink in the curve here. Does the death of a pet prevent its growth until it is ressurected then? This would seem a sensible hypothesis so let us test it by looking at other points on the graph at which the pet was dead. Our original data indicates death during levels 18, 21 and 24 but there is no observable anomaly at these points on the growth curve, as it continues to follow the smooth power trendline. You will notice that the point at day 35 which raised this discussion was not a lone kink, it was preceded by quite a few points lying below the trendline, so it seems highly likely that the anomalies were caused during the data collection process; the observer must have missed the exact points at which the pet level incremented, or more likely, rounding errors are to blame as only integer numbers of days were recorded but this was not the case for the Solar Bear which probably produced a smoother growth trendline as a result. This problem is important to note because it will probably turn up in the other data sets collected by the same person. We have however uncovered a strange fact though: death does not affect the rate of pet growth.

Now we should look at the actual equation: L = 2.03t0.511 which is startlingly close to the L = 1.98t0.521 that we found for the Solar Bear. It might be said that errors in data collection should be to blame for the discrepancy and that both pet species have the same growth curve. What we really want to know though, to make the result relevant to our testing of hypothesis 1, is the range of hero levels at which the pet can be gained, for which we need to consult their respective wiki articles. The current ranges recorded are 50-65 for the Double Dragon and 40-54 for the Solar bear. These ranges overlap, making it more uncertain as to whether they are actually two different ranges. It therefore seems likely that if hypthesis 1 is correct, these two pets are in the same 'growth rate band' so we need the data from a pet which can be gained at a much lower hero level if we want to get to the bottom of this.

The pet species with the second most data from this set is the Stripeless zebra. The wiki article for the Stripeless zebra gives an estimate of 26-39 for the range of hero levels it is available so we shall hope that this is different enough from the other two pet species to give us something helpful. The graph is shown below.

We still get a power curve, but a much messier one with the equation L = 1.42t0.557. The growth exponent is still quite similar to what we saw before, but the coefficient is quite different, although keep in mind that this is just our general impression from looking at the values and that no statistical tests have been conducted to give evidence for this. More importantly, also keep in mind that with a much smaller number of data points, the uncertainty of the trendline is greatly increased which probably somewhat accounts for the untidyness. It is at this point that we might start to think about another variable in the general pet level equation; perhaps it is the threshold level a hero must reach to be able to gain a given pet. Before we do that it would be wise to figure out the growth curves of other species then superimpose them on one graph, which is a problem at the moment because the number of data points for each pet species in the rest of the data we have right now is very small. Let's have a go anyway because there are three left to do which have just enough data points for us to get something that might be meaningful.

First up: The Firefox, range of hero levels recorded as 18-29. Note that the missing data point for the 14th level is because the data collector missed the level change.

Now for the Significant otter, range of hero levels recorded as 18-30 so we should expect something similar to the Firefox.

Last of all, the Ninja tortoise, range of hero levels recorded as 18-30 again. Just an observation: 1.8 is 18/10 as a fraction, and the threshold hero level for these pets is about 18. May be important, but probably isn't.

Now to superimpose all of the known growth curves on one graph.

It is clear that there is a correlation between level range and growth rate. Interestingly, the Firefox looks as though it should have a higher level range than the Stripeless Zebra although from the recorded values we have, the Firefox has the lower range. We can therefore predict that the true level ranges are different (as opposed to the level ranges we have assumed are true but are in reality estimated by Godwiki users and so are not to be relied on too much). This is nice because it means that we have made a testable prediction with which we can test everything we have hypothesised. However, we cannot do this if nobody bothers to find out what the range of a Firefox actually is. Luckily, they have done so in Russian Godville and they give the range as 34-49 (but strangely on the actual Firefox article says that it is tameable at level 30) which fits nicely with what we predicted: the Firefox has a higher level range than the Stripeless Zebra. Let's ignore this little observation though because we don't know if a Russian Firefox is even equivalent to an English one, and we no reason to suppose that it is.

Now after fiddling around with the superimposed curves, it was found that actually, if you kept the exponent at 0.52 (what it seems to roughly be for all of those curves), by just changing the value of the coefficient you can still get all of the growth curves. The same person who plotted those graphs did a little maths and came up with the following:

Solar Bear L = 1.98t0.521L = 1.94t0.52

Double Dragon L = 2.03t0.511L = 1.96t0.52

Firefox L = 1.84t0.515L = 1.80t0.52

Ninja Tortoise L = 2.00t0.463L = 1.75t0.52

The following equations were worked out from two species whose growth curves are not shown above because they are so unreliable. They should be treated cautiously.

Stripeless Zebra L = 1.42t0.557L = 1.62t0.52

Vogon Poet L = 1.01t0.658L = 1.50t0.52 very uncertain

Of course it may be a happy coincidence that you can do this, but there's not much else we can do now until we know the exact threshold hero level for each pet. A way to gather this information would be to write a script to trawl the pets of Godville for their respective hero level ranges, but nobody has bothered to do this yet so we'll have to wait. For now, if we want, we can lie and say that we now know the growth of any pet follows the equation L = kt0.52 where k is the growth factor of that pet species, which correlates with the threshold level a hero must be to tame that pet species, and the higher that growth factor is, the quicker the pet will level up.