Number Choosing Contest.

54345

Woah there!! Another close one with the number rules. What are you even trying?

Python and a hunch. Trouble is, I still don't know what rule 1 is, and I haven't the greatest understanding of the formatting rules, soooooo...

sqrt(121)

3.48757 x 10^5

(sum of the first twenty natural numbers) - (sum of the first thirteen natural numbers)

11: Breaks
1 2 3 5

348757: Breaks
1 2 5

119: Breaks
5 (Close again!!)

54345

Breaks
1 2 3 5 R

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...huh. That's strange.

sqrt(196)

sqrt(7056)

sqrt(8100)

sqrt(9216)

Breaks
1 2 3 5 R U

...huh. That's strange.

sqrt(196)

sqrt(7056)

sqrt(8100)

sqrt(9216)

14: Breaks
2 3 5 U

84: Breaks
1 2 3 5 U

90: Breaks
1 2 3 5 U

96: Breaks
1 2 3 5 U

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Breaks
2 3 5

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211: three-digit prime number with two ones in a row, ending in a one.
311: same
811: same
911: same

C'mon, folks. You've got plenty numbers which satisfied the rules which you're having trouble with.
There's also no real reason to keep putting in numbers which you know break some rule. Process of elimination, right?

Help each other out or something. It's fine to talk about what you think are the rules.


223: Breaks
1 2 3 5 R U
Passes
4 V

224: Breaks
1 2 3 5 R U
Passes
4 V

211: three-digit prime number with two ones in a row, ending in a one.
311: same
811: same
911: same

211:
Breaks
2 3 5 R U
Passes
1 4 V

311:
Breaks
2 3 5 R U
Passes
1 4 V

811:
Breaks
1 2 3 5 R U
Passes
4 V

911:
Breaks
1 2 3 5 R U
Passes
4 V

I mean, I could theoretically just start posting numbers that conform to the three rules I think I've got. There are not that many of reasonable size, so it probably would not take too long to find one that passes everything.

(on that note, you haven't indicated whether or not 811 and 911 pass rule 3 - it is my understanding that they do not)

You are correct - I have updated the previous post appropriately.

It is easier, also, to just say what you suppose the rules are. Brute-forcing is also permissible but less preferred.

I am frustrated at 100 and rule 5.

These numbers passed rule 1:

SpoilerShow

121
123454321
119
120
1
-4
4
16
49284
3364928064
-85
85
13
14
95
111
211
311
1000
1061
123321

These numbers passed rule 2:

SpoilerShow

121
123454321
331130809
11108889
119
120
131160

These numbers passed rule 3:

SpoilerShow

119
1
-85
85
331130809
0
9
3
5
7
348757

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

Guesses:

1849

1369

361

3721

400

It's worth noting that -4 also passed rule 5, though to be honest given how out of my way I went to get 331130809 I think "perfect squares, bar possible mistakes" is the de facto rule 5.

Rule 3, if you want to spoil yourself:

SpoilerShow

Must be a binary palindrome (e.g. 85 = 1010101 or 331130809 = 10011101111001010011110111001)

(this is why I had to get Python involved; to find numbers that pass both rules 3 and 5)

(yes I did get that from just 0, 1, 3, 5, 7, 9 & 85)

(sorry)

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

Rule 3 talk:

SpoilerShow

When you mentioned python, I did think "Do we need to check other bases?" but I dismissed it, mainly due to the apparent straightforwardness of the other two rules. Thank you for doing that grunt work.

It's worth mentioning that rule 1 is also not large-number friendly.

And that rule 2, possibly, is prime-exclusive.

Edit: Number Ninja'ed.

The thing that got a "that's strange" out of me earlier is that rule 1 compliant numbers seemed to come in clumps: {1}, {4}, {13, 14, ??, 16}, {85, ... ,95}, {119, 120, 121}. That none of 84, 90 and 96 passed it irks me something fierce. It'll be interesting actually to see if 15 works...

e

-1/3

17

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There is likely only one correct number below 10^14.

He says, looking at his current shortlist.

("short" is appropriate)

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

Actually, numbers that are in rule one may have something to do with the number of zeroes in binary. (16 has four zeroes, all others have three or less). Can somebody test some powers of two?

There is likely only one correct number below 10^14.

He says, looking at his current shortlist.

("short" is appropriate)

I'd suggest that, if your short list is less than 5 numbers long, then given the stipulations of the game, you've effectively solved the rules here.

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