Number Choosing Contest.

I am frustrated at 100 and rule 5.

[numerals removed for brevity]

Rule 4 is natural numbers greater than zero.
Rule 5 is effectively perfect squares except 100.

I suppose exceptions are a good way to mislead.

But you're not wrong.

Guesses:

1849

1369

361

3721

400

1849: Breaks
2 3 R U

1369: Breaks
1 2 3 R U

361: Breaks
2 3 R U

3721: Breaks
1 2 3 R U

400: Breaks
2 3 R U

It's worth noting that Kaynato was able to tell us that the largest prime broke rule 2, but computation was a limiting factor in determining its status with respect to rules 1 and 3. This suggests that rule 1 is also not an immediately obvious property of a number, whilst 2 probably is.

EDIT: actually since I'm wasting my posts I may as well ask for the gaps at the start of my spreadsheet to be filled:

12

15

Takes a while and I have some resources but I don't think I can handle a number that large without getting into number theory more than I need to right now.

12: Breaks
2 3 5 R U

15: Breaks
1 2 5 R U

e

-1/3

17

e: Did you mean - "Euler's constant?"
Results are displayed:
Breaks
Rule 1: Indefinite. Failed.
Rule 2: Failed.
Rule 3: Indefinite. Failed.
Rule 4: Failed.
Rule 5: Failed.
Breaks R U

-1/3
Rule 1: Failed.
Rule 2: Failed.
Rule 3: Indefinite. Failed.
Rule 4: Failed.
Rule 5: Failed.
Breaks R U

17: Breaks
1 2 5 R U

He's going way more complicated than I expected. I like this.

Since if we're working together we can't really have the winner become the new leader, should we elect the next one?

Testing for rule one:
-121
-123454321
-119
-120
-1

Sure, election's fine, once someone gets it.

-121: Didn't we do these a while ago?
Breaks:
3 4 R U

-123454321: Breaks
3 4 R U

-119: Breaks
4 5 R U

-120: Breaks
3 4 5 R U

-1: Breaks
2 4 R U

Not necessarily. If we were only told whether it passed ALL the rules, it would be a little unfair. However, we know which pass which rules individually, and its only a problem to combine them. Once we figure out the rules, it will be easy to solve for a satisfying solution, no matter how large it is.

also:
2
4
16
256
65636

Good way to put it.

Breaks rule 1: 2, 65636 (I get the feeling this wasn't the number you wanted)
Breaks rule 2: 2, 4, 16, 256, 65636
Breaks rule 3: All of these.
Breaks rule 4: None of these.
Breaks rule 5: 65636.

Breaks rule R: All
Breaks rule U: All
Breaks rule V: None