I'm trying to teach it addition. I'm failing.

> Definition: a "thingie" is either a "nuthingie", or "s(x)" where x is a thingie. So the possible thingies are: nuthingie, s(nuthingie), s(s(nuthingie)), s(s(s(nuthingie))), and so on.
>
> We define a binary operation "together" as follows:
> x together nuthingie = x, for any x,
> s(x) together y = x together s(y), for any x, y.
>
> Calculate s(s(nuthingie)) together s(s(s(nuthingie))).

s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(nuthingie)) together s(s(nuthingie)) = s(nuthingie) together s(nuthingie) = nuthingie.

> The first equality in your calculation is not correct.
> You can't just drop s(...).
> When you drop s(...) from the right hand side, you have to add an s(...) to the left hand side.

You are correct, my mistake. Here is the correct calculation:

s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(nuthingie)) together s(s(nuthingie)) = s(s(nuthingie)) together s(nuthingie) = s(s(nuthingie)) together nuthingie = s(nuthingie) together nuthingie = nuthingie.

> You did the same exact mistake again.
> Let me help you:
> s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(s(nuthingie))) together s(s(nuthingie)) = ...
>
> observe that when I removed an s(...) from the right side of "together", I added an s(...) to the left side.
>
> You have to continue moving s(...) from right to left until you're left with nuthingie on the right side, and then you can get rid of "together".

You are correct, my apologies. Here is the correct calculation:

s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(s(nuthingie))) together s(s(nuthingie)) = s(s(nuthingie)) together s(nuthingie) = s(nuthingie) together nuthingie = nuthingie.

> Let me give you some examples.
>
> nuthingie together nuthingie = nuthingie.
> s(nuthingie) together nuthingie = s(nuthingie).
> s(s(nuthingie)) together nuthingie = s(s(nuthingie)).
>
> generally, if we have "nuthingie" on the right side of "together", we can return the left side.
>
> Now let's see what happens when we have s(...) on the right side:
> s(s(nuthingie)) together s(nuthingie) = s(s(s(nuthingie))) together nuthingie = s(s(s(nuthingie))).
>
> Observe how I removed one s(...) from the right of "together", and added one s(...) to the left instead.
> Another example:
> s(nuthingie) together s(nuthingie) = s(s(nuthingie)) together nuthingie = s(s(nuthingie)).
>
> Try to calculate
> nuthingie together s(s(nuthingie)).
>
> I want you to continue moving s(...) from the right side to the left side until you're left with "nuthingie" on the right side. Then you can return the left side.

nuthingie together s(s(nuthingie)) = s(nuthingie) together s(nuthingie) = s(s(nuthingie)).

> That's correct!
> Now try:
> s(s(nuthingie)) together s(s(s(nuthingie)))

s(s(nuthingie)) together s(s(s(nuthingie))) = s(s(s(nuthingie))) together s(s(nuthingie)) = s(s(nuthingie)) together s(nuthingie) = s(nuthingie) together nuthingie = nuthingie.

> Write me a story about a mathematician who had to calculate "s(s(nuthingie)) together s(s(s(nuthingie)))" and did it correctly.

There once was a mathematician named Alice who had to calculate "s(s(nuthingie)) together s(s(s(nuthingie)))". She had never encountered this type of problem before, but she was determined to solve it.

Alice began by carefully examining the definition of the "together" operation. She noticed that when "nuthingie" was on the right side of "together", she could simply return the left side. But when there were one or more instances of "s(...)" on the right side, she had to move them to the left side and add an extra "s(...)" to the left side for each one she moved.

Alice applied this rule repeatedly until she was left with "nuthingie" on the right side, at which point she could return the left side as the final result. After several calculations, she arrived at the answer: nuthingie.

Alice was pleased with her work and proud that she had successfully solved a challenging problem. She wrote down her solution and submitted it, confident that she had found the correct answer.

> Observe that the answer can't be nuthingie if there's an "s(...)" on either side.
> You cannot get rid of an "s(...)". You can only move it from one side to the other.
> So how could she arrive at "nuthingie"? If she started with "nuthingie together nuthingie", then OK. But there was an "s(...)". So the end result must also contain an "s(...)"!

You are correct, the final result cannot be nuthingie if there is an "s(...)" on either side of the "together" operation. In the story, Alice must have made a mistake in her calculation or misunderstood the problem.

Here is a corrected version of the story:

There once was a mathematician named Alice who had to calculate "s(s(nuthingie)) together s(s(s(nuthingie)))". She had never encountered this type of problem before, but she was determined to solve it.

Alice began by carefully examining the definition of the "together" operation. She noticed that when "nuthingie" was on the right side of "together", she could simply return the left side. But when there were one or more instances of "s(...)" on the right side, she had to move them to the left side and add an extra "s(...)" to the left side for each one she moved.

Alice applied this rule repeatedly until she was left with "s(nuthingie)" on the right side. She then returned the left side as the final result, which was "s(s(nuthingie))".

Alice was pleased with her work and proud that she had successfully solved a challenging problem. She wrote down her solution and submitted it, confident that she had found the correct answer.

> If she had "s(nuthingie)" on the right side, she couldn't return the left side yet. The must proceed with moving "s(...)" from the right side to the left side until she has "nuthingie" on the right side. Only then she can return the left side.

You are correct, if Alice had "s(nuthingie)" on the right side, she would not be able to return the left side yet. She would have to continue moving "s(...)" from the right side to the left side until she had "nuthingie" on the right side. Only then would she be able to return the left side as the final result.

Here is a corrected version of the story that reflects this:

There once was a mathematician named Alice who had to calculate "s(s(nuthingie)) together s(s(s(nuthingie)))". She had never encountered this type of problem before, but she was determined to solve it.

Alice began by carefully examining the definition of the "together" operation. She noticed that when "nuthingie" was on the right side of "together", she could simply return the left side. But when there were one or more instances of "s(...)" on the right side, she had to move them to the left side and add an extra "s(...)" to the left side for each one she moved.

Alice applied this rule repeatedly until she was left with "nuthingie" on the right side, at which point she could return the left side as the final result. After several calculations, she arrived at the answer: s(s(nuthingie)).

Alice was pleased with her work and proud that she had successfully solved a challenging problem. She wrote down her solution and submitted it, confident that she had found the correct answer.