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This is in the spirit of the What is a Word/Phrase™ series started by JLee with Number version puzzles.

If a number conforms to a special rule, I call it a BEN Number™.
Use the following examples below to find the rule.

$$ % set Title text. (spaces around the text ARE important; do not remove.) % increase Pad value only if your entries are longer than the title bar. % \def\Pad{\P{0.0}} \def\Title{\textbf{ BEN }} % \def\S#1#2{\Space{#1}{20px}{#2px}}\def\P#1{\V{#1em}}\def\V#1{\S{#1}{9}} \def\T{\Title\textbf{Numbers}^{\;\!™}\Pad}\def\NT{\Pad\textbf{Not}\T\ }\displaystyle \smash{\lower{29px}\bbox[yellow]{\phantom{\rlap{rubio.2019.05.15}\S{6px}{0} \begin{array}{cc}\Pad\T&\NT\\\end{array}}}}\atop\def\V#1{\S{#1}{5}} \begin{array}{|c|c|}\hline\Pad\T&\NT\\\hline % \text{ 17 }&\text{ 57 }\\ \hline \text{ 20 }&\text{ 60 }\\ \hline \text{ 24 }&\text{ 64 }\\ \hline \text{ 49 }&\text{ 89 }\\ \hline \text{ 121 }&\text{ 161 }\\ \hline \text{ 158 }&\text{ 198 }\\ \hline \text{ 538 }&\text{ 578 }\\ \hline \text{ 830 }&\text{ 870 }\\ \hline \text{ 1,059 }&\text{ 1,099 }\\ \hline \text{ 1,144 }&\text{ 1,184 }\\ \hline \text{ 1,622 }&\text{ 1,662 }\\ \hline \text{ 2,345 }&\text{ 2,385 }\\ \hline \end{array}$$

And, if you want to analyze, here is a CSV version:

BEN Numbers™,Not BEN Numbers™
17,57
20,60
24,64
49,89
121,161
158,198
538,578
830,870
1059,1099
1144,1184
1622,1662
2345,2385

Bonus Question:
It's true that the cardinality of BEN Numbers' set is finite. What is the largest BEN Number?


Hint 1:

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Hint 2:

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Hint 3:

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Hint 4:

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Hint 5:

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Hint 6:

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Hint 7:

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Conifers
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    Are the examples of BEN numbers and non-BEN numbers ordered sequentially, or are these just arbitrary examples? In other words, could there be another BEN number between 17 and 24 or 24 and 49, for instance? – SparkFountain Sep 10 '19 at 14:53
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    Yes, there are still many BEN Numbers not mentioned in the question. And it's true that another BEN Number exists in the 17 ~ 24 & 24 ~ 49. – Conifers Sep 10 '19 at 14:59
  • Can you provide a hint? – Abhi Sep 11 '19 at 09:13
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    Here are my thoughts which might help someone solve this (because I couldn't): rot13(V abgvprq gung gur ahzoref ba gur yrsg arire raq jvgu 60..99, fb V gubhtug gur yrsg pbyhza jrer nyy gvzrf sebz 00:00 gb 23:59. Gur evtug pbyhza unf ahzoref gung nyy raq va 60..99, rkprcg sbe gur svefg ahzore 57 juvpu V pna'g rkcynva. Nyfb "ORA" pbhyq ersre gb "Ovt Ora". Gur svefg uvag fubjf "Fha Zbba Erfreibve", naq gur frpbaq fubjf qnlyvtug uvggvat gur rnegu, fb gur nafjre zhfg unir fbzrguvat gb qb jvgu gung naq abg whfg gvzr.) – JS1 Sep 12 '19 at 21:20
  • @JS1 You are on the way :P. Think more deeply about the "BEN"~ – Conifers Sep 13 '19 at 02:07
  • I have a feeling the hints may suggest something about rot13(rpyvcfrf), am I too far? – Pretzel Sep 17 '19 at 10:50
  • @VictorPerszel Yeh that's way too far :P – Conifers Sep 17 '19 at 10:52
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    I think it could have something to do with rot(n pybpx-snpr. Vs ORA eryngrf gb gur Ovt Ora jngpugbjre. Gur ORA ahzoref pbhyq or genafyngrq gb jngpu unaqf naq fbzr qrterr (uvag4) orgjrra gur gevnatyr gurl perngr (uvag3) pbhyq or gur xrl.) – 53RT Sep 17 '19 at 12:58
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    Does it have something to do with rot13(gvzr) or more specifically a rot13(fha qvny)? This would somehow match to the hints. – 53RT Sep 18 '19 at 15:29
  • @53RT maybe part of~ – Conifers Sep 18 '19 at 16:35
  • I think I've now done more research on Big Ben than would be considered healthy :P – Dmihawk Sep 19 '19 at 20:55
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    Has anyone noticed that rot13( ORA naq abg ORA ahzoref qvssre ol 40. V guvax gur BC chgf guvf vagragvbanyyl. ) – Luke Sep 20 '19 at 00:57
  • @Luke 40 is arbitrarily I chose the number :P(But yes 40 could be counted as a little hint I think~, not so correlated comparing to those above 6 hints) – Conifers Sep 23 '19 at 08:29
  • rot13( Fb, vg frrzf yvxr Uvag #7 vf gryyvat hf gb "fcyvg" gur ahzoref. Gurl pna or fcyvg vagb gvzrf ba n 24 ubhe pybpx yvxr 23:45, ohg 00:57 jbhyq or n inyvq ahzore nyfb). – GrumpyLlama59 Sep 23 '19 at 20:20
  • Is hint #5 for real? I have no idea what it means – GrumpyLlama59 Sep 23 '19 at 20:34
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    New thoughts: rot13(Uvag #1'f ybpngvba vf pybfr gb TCF pbbeqvangrf "24.0, 120.0". Creuncf guvf unf gb qb jvgu n "fcyvg" (uvag #7) bs gur ahzoref vagb [0..23][0..119]. Uvagf 2/3/4, znl cbvag gb trbpbbeqvangrf, natyrf, naq qrterrf. V fcyvg gur ahzoref naq znccrq gur [0..23] cneg nf ybatvghqr ahzoref, ol zhygvcylvat gur ahzore ol 15 gb znxr gur enatr [0..359]. V gerngrq gur frpbaq cneg nf n yngvghqr ahzore (abegu). Ohg guvf qvqa'g tvir nalguvat hfrshy. Vg znl arrq n qvssrerag vagrecergngvba, rfcrpvnyyl sbe gur 2aq cneg. Nyfb uvagf 5/6 ner haxabja. Png/Qbt = P/Q?) – JS1 Sep 23 '19 at 22:08
  • rot13(Va gur pnfr gung n Ora ahzore vf ersreevat gb Ovt Ora juvpu unf 4 pybpx snprf, jr zvtug arrq gb qvivqr ol 4) – GrumpyLlama59 Sep 23 '19 at 22:40
  • rot13(Sbe uvag 5, V pbhyq nqq zber vzntrf yvxr enoovg be qrre. Fbzrbar va pbzzragf zvtug or va gur evtug jnl gb fbyir~) – Conifers Sep 24 '19 at 00:57
  • Well, seems I'm about to be out of new hints to give... Maybe still 1 or 2 more :P – Conifers Sep 24 '19 at 01:02
  • Any scope of adding another pair or two of BEN/not BEN numbers to the table? Are there single digit BEN numbers? – Certainly not a dog Sep 24 '19 at 10:29
  • @Certainlynotadog Yes, it's true that there has no any single digit BEN Numbers :D – Conifers Sep 24 '19 at 11:05
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    About Hint #5 rot13(Gur chccl naq gur xvggl ner zrnag gb or 'phgr' naq zl gubhtugf vf gung vg vf orpnhfr bs gur nphgr natyr, eryngvat nyfb gb bgure uvagf (:) – Pretzel Sep 24 '19 at 11:46

1 Answers1

17

Ok, so a Ben number is a number that

when split like a clock number, and put on a clock, make an acute angle (< 90º)

Proof:

17 -> 01:07 -> 12º
20 -> 02:00 -> 60º
24 -> 02:04 -> 36º
49 -> 04:09 -> 66º
121 -> 01:21 -> 66º
158 -> 01:58 -> 42º
538 -> 05:38 -> 78º
830 -> 08:30 -> 60º
1059 -> 10:59 -> 54º
1144 -> 11:44 -> 66º
1622 -> 16:22 -> 12º
2345 -> 23:45 -> 60º
(To calculate just multiply the hour by 5, subtract the minutes, and then the multiply by 6)

All numbers in the Not Ben Column end up with a number bigger than 59 (except the first 57 -> 05:07 -> 108º)

The hints helped so:

1-

It's the Sun Moon Reservoir, sun and moon indicating time (thanks to @JS1)

2-

It's representing day

3-

A triangle, that contains 3 acute angles

4-

Degree

5-

They are cute -> Acute

6-

BEN -> Big Ben, the clock

7- What helped me

From the movie 'SPLIT', this is what helped me because I was splitting the numbers differently (I was making 49 as 00:49, not 04:09)

The largest BEN number is then

2359 -> 23:59 -> 24º

PiGuy314
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Pretzel
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