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escape the room! 6
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by
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dustin
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NESDORI                           2021-09-15 23:07
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dustin, EXTRA PUZZLE FOR YOU: HOW MANY NUMBERS THERE'LL BE IN THAT SEQUENCE UNTIL THERE'S A FIRST NUMBER WITH 3 DIGITS? :)

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DUSTIN                            2021-09-16 00:46
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OH MY... THAT SOUNDS HARD :D

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DUSTIN                            2021-09-19 11:09
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HERE'S THE SOLUTION (SPOILER WARNING IF YOU STILL WANT TO TRY):  
  
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THE N-TH NUMBER IN THE SERIES IS THE DIGIT SUM OF N SQUARED.  
X IS THE EIGHTH NUMBER, SO THE DIGIT SUM OF 64 IS 10.  
Y IS THE 14TH NUMBER. THE DIGIT SUM OF 196 IS 16.  
SO THE HIDDEN EXIT IS AT (10, 16)! ;D  
  
nesdori, I STILL HAVE NO CLUE OTHER THAN BRUTE FORCE :D

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DUSTIN                            2022-08-05 22:42
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nesdori, I'VE THOUGHT ABOUT YOUR EXTRA PUZZLE A FEW TIMES WITHOUT SUCCESS. I'M CURIOUS FOR THE ANSWER! :D

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NESDORI                           2022-08-18 13:39
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dustin, THE FIRST NUMBER WITH A 3-DIGIT SQUARE DIGIT SUM PUZZLE IS INDEED TOUGH ALTHOUGH THE SEARCH SPACE CAN BE REDUCED. I WAS JUST CURIOUS HOW A MATH WIZ LIKE YOU WILL DEAL WITH IT :)  
  
THE LOWEST POSSIBLE SQUARE WITH DIGIT SUM OF AT LEAST 100 IS 199999999999 SO THE NUMBER WE ARE LOOKING FOR IS AT LEAST  
CEIL(SQRT(199999999999) = 447214. IF THE SQUARE ROOT RESULT WOULD HAVE BEEN AN INTEGER WE ALREADY FOUND THE ANSWER BUT UNFORTUNATELY NO LUCK :)  
  
THE NEXT POSSIBLE RANGE OF SQUARES IS 289999999999 - 299999999999 (538517 - 547722) BUT TO INSPECT THIS RANGE ONLY 12 INTEGER-RESULT CHECKS ARE NEEDED (2 FOLLOWING WITH ALL NINES AND WITH EACH 9 REPLACED WITH 8). NUMBERS IN THE RANGE 447215 - 538516 ARE NOT POSSIBLE ANSWERS AND CAN BE JUST SKIPPED.  
  
THE NEXT SQUARE RANGE IS 379999999999 - 399999999999 (616442 - 632455, 547723 - 616441 CAN BE AGAIN SKIPPED) AND ALREADY HAS 78 NUMBERS TO CHECK. SO IT'S ALREADY BECOMING HARD TO CHECK RANGES BY HAND. AT SOME POINT THERE ARE TOO MANY CHECKS TO BE DONE USING THIS METHOD AND TRADITIONAL BRUTE-FORCING BECOMES EASIER.  
  
THE NUMBER WE ARE LOOKING WAS ALMOST FOUND BEFORE REACHING 1M. THE SQUARE DIGIT SUM OF 987917 IS 97.  THE FIRST NUMBER WITH A SQUARE SUM OF 100 OR BIGGER IS 1643167 (SQUARE IS 2699997789889, DIGIT SUM EXACTLY 100).  
  
BUT I DIDN'T STOP THERE :D I WAS WONDERING THAT WHAT MIGHT BE THE FIRST NUMBER WITH A SQUARE DIGIT SUM OF AT LEAST 200. THE NUMBER IS 3144835432874 WHICH SQUARED IS 9889989899859798959899876 (SUM 202). FINDING THIS NUMBER TOOK A LOT OF COMPUTATION TIME (A FEW DAYS). BECAUSE THE SQUARES ALREADY ARE BIGGER THAN 64-BIT VALUE LIMIT IT'S A BIT TROUBLESOME TO CALCULATE THOSE BY A COMPUTER WITH 64-BIT ARCHITECTURE :)

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DUSTIN                            2022-08-24 18:33
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nesdori, AH! I THOUGHT THERE WAS AN ELEGANT MATHEMATICAL TRICK TO SOLVE THIS :D PROGRAMMING IS NOT MY STRENGTH, SO I DIDN'T REALLY HAVE A CHANCE ON THAT ONE :D  STILL A NICE RESULT! :D