Recently the following question has been circling the Internet and also ended up in my inbox a few times. It's yet another one of those silly exercises that ask you to continue a sequence based on a number of examples. They are a pet peeve of mine and I'll explain why I hate them here once and for all instead of replying to everyone that sent a copy my way.
The logic of this specimen is particularly broken, even putting aside the questionable statement about pre-school children being able to solve it in mere minutes. It's obvious that whoever wrote that is not familiar with the meaning of the equals sign, as 8809 isn't equal to 6. So strictly reading, this exercise gives you 21 false statements. From that it can be simply inferred that the author is looking for another false statement, so ??? can be replaced with any number not equal to 2581.
With that out of the way, let's look at what was probably meant with this notation. You are given N=21 pairs of values (xi, yi) such that:
And asked to find yN for a given xN:
From a purely mathematical standpoint, this problem has an infinite number of solutions. To demonstrate, assume that f is a function that satisfies the condition set above. In that case:
also satisfies this condition. Note that g is a function that can have an arbitrary value at xN and hence this new solution will give a different value of yN.
Just to make this a little bit more practical, here's an example of a polynomial that goes through N points:
And here is how its graph looks like if you plug in the given 21 values:
At x=2581 it gives the value of around 5.8·1073.
Note that using the formula above and N=22, you can get a polynomial that goes through the required 21 points plus the 22nd that you can choose to your liking.
This is just one example. You could construct functions that meet the requirements from trigonometric or exponential functions in a similar way.
Sure, so this is probably not the solution the author of this question was looking for. You could say that I'm overcomplicating things and that the rule that pre-school children would come up with, based on the shape of the digits the author arbitrary chose to represent these numbers in, is nicer. But there is no such thing as niceness in mathematics (see interesting number paradox).
This is the big difference between mathematics and physics. Our universe has physical laws that for some reason can often be represented in certain mathematical forms. If a liter of water has a mass of one kilogram and two liters a mass of two kilograms it's quite likely that three liters will have a mass of three kilograms and so on. But this is purely based on our experience and beliefs about our environment and not on any purely mathematical basis. Such a prediction, just like the one above and many like it, when stripped of any physical context, does not make any sense.