Number Choosing Contest.

It is six numbers long.

@a52: 111 passes rule 1 and has binary expression 1101111; rule 5 fails rule 1 and has binary expression 101.

how does a rule fail another rule

@Sruixan
Yeah, good point.

how does a rule fail another rule

My brain fails rule 6, which is "make sense at all times".

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

I've noticed numbers ending in non-zero even numbers tend to not pass Rule2?

2636
1084674
28672
209678

30560

I've noticed numbers ending in non-zero even numbers tend to not pass Rule2?

2636
1084674
28672
209678

30560

Breaks...
1: 1084674, 28672, 30560
2: 2636, 1084674, 28672, 30560
3: All
4: None
5: All
R: All
U: All
V: None

imaginary 420
72
99
44100
45369

imaginary 420
72
99
44100
45369

Breaks...
1: 420i,
2: All except 45369
3: All except 99
4: 420i
5: 420i, 72, 99
R: All except 420i
U: All except 420i
V: None

100+99i

100+99i

1: Invalid. Failed.
2: Invalid. Failed.
3: Invalid. Failed.
4: Failed.
5: Failed.
Passes format rules.

passes format rules, interesting
100 + 21
100 + 19
91 + 6x, where x = 5.

The Glaisher–Kinkelin constant A

Graham's number

passes format rules, interesting
100 + 21
100 + 19
91 + 6x, where x = 5.

Oh, fun times.

121: breaks 3 V
119: breaks 5 U V
121: breaks 3 V

The Glaisher–Kinkelin constant A

Breaks all number rules, passes format rules.

Graham's number

Come on.

Rules 1, 2, 3, 5 inconclusive. Effective failure.
Passes rule 4 and format rules.
Counts as no-pass. Success must be certain.

eighty six
80 seven
eighty 8
eighty-nine
four score and ten

266
-266i

eighty six
80 seven
eighty 8
eighty-nine
four score and ten

Breaks...
1: All
2: All
3: All
4: None
5: All
R: None
U: None
V: None

266
-266i

266: Breaks
1 2 3 5 R U

-266i: Breaks
1 2 3 4 5 R U

2^57,885,161 : (Largest known prime +1)

4000

202

18

1388741444313152705354935065649 : Largest number I could find that should pass 3 and 5.

2^57,885,161 : (Largest known prime +1)

4000

202

18

1388741444313152705354935065649 : Largest number I could find that should pass 3 and 5.

2^57,885,161
Rule 1: Computational resources not enough. Effective failure.
Rule 2: Breaks.
Rule 3: Breaks.
Rule 4: Passes.
Rule 5: Breaks.

4000: Breaks
2 3 5 R U

202: Breaks
2 3 5 R U

18: Breaks
1 2 3 5 R U

1388741444313152705354935065649:
Rule 1: Passes.
Rule 2: Passes.
Rule 3: Passes.
Rule 4: Passes.
Rule 5: Passes.
Breaks R U

holy shiT!
1 nonillion 388 octillion 741 septillion 444 sextillion 313 quintillion 152 quadrillion 705 trillion 354 billion 935 million 65 thousand 649

Welp. Never mind.

Just for the sake of closure: my list consisted of

2.0457529 × 10^7

1.43784081 × 10^8

2.0074072489 x 10^9

1.193532215121 x 10^12

1.0036851273801 x 10^13

4.1413201925481 x 10^13

...whose square roots you can see in bob's list. It was the OEIS that I used to get rule 3, for the record.

okay so that passed all the number rules.

Is the goal of the game to find a number that meets the rules or is it to figure out what the rules are?

I'll format the top one for the former goal and submit a few more guesses for the latter.

1.388741444313152705354935065649 x 10^30

2110

1111

31

22