29
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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:

enter image description here

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  • 2
    $\begingroup$ 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? $\endgroup$ – Spark Fountain Sep 10 at 14:53
  • 1
    $\begingroup$ 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. $\endgroup$ – Conifers Sep 10 at 14:59
  • 13
    $\begingroup$ 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.) $\endgroup$ – JS1 Sep 12 at 21:20
  • 1
    $\begingroup$ 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.) $\endgroup$ – 53RT Sep 17 at 12:58
  • 1
    $\begingroup$ Does it have something to do with rot13(gvzr) or more specifically a rot13(fha qvny)? This would somehow match to the hints. $\endgroup$ – 53RT Sep 18 at 15:29
14
+50
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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 represanting 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

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  • $\begingroup$ This looks correct! Nice answer and good catch on #5! $\endgroup$ – Zoir Sep 24 at 12:16
  • 1
    $\begingroup$ good one. A little thing, the hour hand usually moves during the minutes, so it is at 8:30 between 8 and 9. But the angle are still acute. $\endgroup$ – 53RT Sep 24 at 12:24
  • $\begingroup$ @53RT Yes, you're right! But as there was no angle between 85º and 95º we're safe here (: $\endgroup$ – Pretzel Sep 24 at 12:31
  • $\begingroup$ YES! Finally someone get it~ well done :P $\endgroup$ – Conifers Sep 24 at 14:35
  • 2
    $\begingroup$ @Conifers thx! I got about the angle at hint #5 but I was splitting the numbers wrong :c Thanks for the nice puzzle! $\endgroup$ – Pretzel Sep 24 at 14:54

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