Monday, July 30, 2018

pattern - What is a BEN Number™?


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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Answer



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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