Number Choosing Contest.

Right, now I need to figure out what rule 4 is...

39

63

95

8191

39:
Breaks Rules 2, 3, 4, 5

63:
Breaks rule 3

95:
Breaks rules 2, 3, 5

8191:
Breaks rule 5

654321

8658

654321:
Breaks rules 2, 3, 5

8658:
Breaks rules 3, 4, 5

223

239

463

911

1231

All of these break rule 5


Breaks rules 1 and 3

SpoilerShow

Mathematica is great.

1583

2143

2239

2351

2591

All of these break rule 5

Kaynato, if I understand what you think rule 4 is correctly, you're going to have to count for quite a while to get something that is also rule 5 compliant.

Up to 33391, to be exact.

Yup, up to exactly that number!
(I'm not answering the rest because it looks like the assumptions you made were probably correct)

1: Number must not contain 7 in base ten
2: Number must be one less than, one greater than, or equal to a multiple of 7
3: Number must be prime
4: Number must contain an F when written in hexadecimal
5: Number must be expressible as the sum or difference of two nonzero cubes

Prerequisites:

SpoilerShow

α: Number must be constant
β: Number must be finite
γ: Number must be real
δ: Number must be an integer
ε: Number must be positive

Looks like I made it a little easy. I'm surprised that you figured out rule 5 before rule 4. When did you figure out rule 5?

Right after I posted that one with the screenshot I figured I'd enter the ones which passed rule 5 into OEIS.

It gave me exactly what was needed.

The way I put together rule 4 was different, actually, but it works out all the same.

This is what I put together:

Looks like I made it a little easy. I'm surprised that you figured out rule 5 before rule 4. When did you figure out rule 5?

Well, a few minutes before I specifically asked for 631, knowing full well it passed rules 1, 2, 3 and 5...

(and yes like Kaynato I did so via OEIS)

Sruixan, are you up to run the next round?

Yep, can do. I'll be going to bed in a minute, so it'll either start if I get time over lunch tomorrow or (more likely) in my evening...

Man, I want Mathematica, but it is so expensive.

There are two numbers below 1,000,000 that pass all five of my chosen rules. These rules have been numbered 1-5 in no particular order, and are joined by four others, W-X-Y-Z, which cover both the presentation of the number and matters of general number-choosing etiquette. Any errors in my judgement of your numbers will be the fault of the Python script by which I am grading them and clearly not mine, despite the fact that it was I who wrote the script in the first place.

Good luck!

-1

0

1

2

3

999999

-1

0

1

2

3

-1: breaks WYZ & 1, 3, 4 and 5.
0: breaks WYZ & 2, 3 and 4. It strictly speaking passes 1 and fails 5, but you could contest me on either.
1: breaks YZ & 3, 4 and 5.
2: breaks YZ & 2, 3, 4 and 5.
3: breaks YZ & 2, 3, 4 and 5.

999999

999999: breaks YZ & 1, 2, 4 and 5.

Four

The sum of two and three

factorial(3)

The fifth positive nonzero prime

Two raised to the third power

Four

The sum of two and three

factorial(3)

The fifth positive nonzero prime

Two raised to the third power

4: breaks XYZ & 2, 3, 4 and 5.
5: breaks XY & 2, 3, 4 and 5.
6: breaks XYZ & 2, 3, 4 and 5.
7: breaks XY & 3, 4 and 5.
8: breaks XY & 2, 3, 4 and 5.

For the record, people are more than welcome to try numbers above a million if they think it'll be informative. You never know!

3^2

5*2

10+1

1100 (base 2)

baker's dozen

3^2

5*2

10+1

1100 (base 2)

baker's dozen

9: breaks XYZ & 2, 3, 4 and 5.
10: breaks XYZ & 4 and 5.
11: breaks XYZ & 2, 4 and 5.
12: breaks XYZ & 2, 4 and 5.
13: breaks XY & 4 and 5.

ONE MILLION

1234567890ABCDEFGHIJKLMNOPQRSTUVWXYZ₃₆

ONE MILLION

1000000: breaks XYZ & 1 and 5.

(thus making it the first number to pass rule 4)

Ackermann(6,199)

1234567890ABCDEFGHIJKLMNOPQRSTUVWXYZ₃₆

*disgruntled sigh*

3126485650002806059265235559620383787531710118313327355: breaks YZ & 1, 2, 3, 4 and (almost certainly) 5 probably 5.

Ackermann(6,199)

I BROUGHT THIS UPON MYSELF

*number omitted for brevity*: breaks XYZ & definitely 1, maybe 2, very certainly 3, I am absolutely not figuring out 4 and almost certainly 5 maybe 5.

hi sruix why do you do this to yourself?

my guess is 12

hi sruix why do you do this to yourself?

my guess is 12

I have no idea!

12: breaks YZ & 2, 4 and 5.

01001001001000000100110001001111010101100100010100100000010110010100111101010101001000000100001001000010010110010010000000111011001010010010100100101001

01001001001000000100110001001111010101100100010100100000010110010100111101010101001000000100001001000010010110010010000000111011001010010010100100101001

1630767959559389054594297182400609699502827817: breaks XYZ & 1, 2 and 3. Having given it more thought, I don't think I can make judgements on rule 5 for numbers this large and will shortly edit my previous post to reflect this. This does pass rule 4, though! You may need some other examples bar this and a million to get anywhere with it, but at least it's something, right?

TWO MILLION

TWO MILLION

2000000: breaks XYZ & 1, 2, 3, 4 and 5.