Comment by tibbar

Curiously, that makes integration a trapdoor function and a possible cryptographic primitive. Not a very practical one, but...

(It looks like OP's most recent post was deleted, but I spent some time writing this up so I'll post it anyway. The deleted post contended that Cleo never confessed to "cheating" by differentiating and then reversing the direction.)

I think you are misunderstanding something that is just implicit in the story. IE he doesn't need to "confess" this, it's a fundamental part of how the trick works.

I'll take one more stab at explaining what's going on. (Out of curiosity, are you familiar with calculus? I don't want to assume that you're not, but your comment reads as if you're not very familiar with it, so I'm going to explain things a bit better this time.)

Let's start with how the beautiful parlor trick looked to everyone else.

Random user: Asks how to integrate ABC expression.

(This is difficult, because integrate(ABC) has no general algorithm. It often requires many subtle tricks to perform a given integration, and there is no guarantee that there even is an elementary answer for integrate(ABC)).

Cleo: Answers integrate(ABC) = XYZ, with no notes.

(Wow! This obviously must have required many subtle tricks, but they are not provided!) Importantly, anyone can verify that Cleo is right, because it turns out that UndoIntegration(XYZ) => ABC is easy, and there is a deterministic algorithm to do it. So, we all can tell that Cleo's answer is correct. But how could she have done this, since the integration direction is difficult???

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How the parlor trick really works.

First, as you can see, if Cleo takes any random XYZ, she can easily run UndoIntegration(XYZ) => ABC, and now she knows for free that Integrate(ABC) => XYZ. The neat thing is that doing this doesn't require figuring out any of the subtle steps required to run the Integrate operation either, which is convenient since Cleo doesn't plan to post them anyway.

So then, how does Cleo find a good ABC and XYZ without being a genius who is smarter than a computer? The most important thing is to find a pair such that ComputerAlgebraSystem_Integrate(ABC) doesn't work. Since integration is generally done by a bag of tricks, there are always holes you can find. So, you can basically just do this:

1. Start with a candidate XYZ_1.

2. Run UndoIntegration(XYZ_1) => ABC_1. (Remember, this is easy.)

3. Check if ComputerAlgebraSystem_Integrate(ABC_1) works. (Also easy to check, although the computer program has to work hard.)

4. If the computer is stumped, good. We can make a StackOverflow post.

5. Otherwise, try a different XYZ_2 and go back to step 1.

This is still an interesting game, but at no point does Cleo need to come up with a crazy bag of integration tricks here, the way that everyone assumes she did when she runs the parlor trick in the forum. This is the whole point of the trick, and the reason she hid her identity.