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arno 1416 the 73-cycle
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by
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Arno
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ARNO                              2020-07-14 22:03
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AS AN ILLUSTRATION, THIS CAVE SHOWS THE FULL MAP WHEN EXTENDED TO 73 COLUMNS, SUCH THAT EACH COLUMN USES THE SAME SEED2 VALUES ON EACH ROW. THE FILLING RULES OF BD1 CAVE A ARE AGAIN USED.  
  
HERE ARE SOME PICTURES SHOWING THE UNDERLYING NUMBERS:  
  
SEED2:  
<a href="http://www.boulderdash.nl/forum/forumpics/73cycle_seed2.png" target="external">HTTP://WWW.BOULDERDASH.NL/FORUM/FORUMPICS/73CYCLE_SEED2.PNG</a>  
  
SEED1:  
<a href="http://www.boulderdash.nl/forum/forumpics/73cycle_seed1.png" target="external">HTTP://WWW.BOULDERDASH.NL/FORUM/FORUMPICS/73CYCLE_SEED1.PNG</a>  
  
INCREMENT OF SEED1:  
<a href="http://www.boulderdash.nl/forum/forumpics/73cycle_seed1_increment.png" target="external">HTTP://WWW.BOULDERDASH.NL/FORUM/FORUMPICS/73CYCLE_SEED1_INCREMENT.PNG</a>  
  
WHILE SEED2 REPEATS EACH ROW, SEED1 STARTS REPEATING ITSELF FROM ROW 14 COLUMN 3. THAT IS, AFTER 952 DRAWS.  
  
AND dustin: YOU NOTICED EARLIER THAT THE CAVE IS 'SILENT' WHICH MEANS THERE IS NEVER EMPTY SPACE DIRECTLY BELOW | OR \.  
THIS CAN BE EASILY EXPLAINED BY THE FORMULA OF SEED1:  
  
S_1 := (S_1 + 128[S_1 ODD] + FLOOR(S_2 / 2) + C_2 + C_3) MOD 256  
  
FOR EACH GIVEN POSITION, THE ITEM BELOW THAT POSITION IS DETERMINED BY EXACTLY 73 DRAWS.   
SINCE S_2 CONTAINS A CYCLE OF 73, OVER 73 DRAWS THE TERMS FLOOR(S_2 / 2) ADD UP TO A CONSTANT VALUE. I CALCULATED IT: IT'S 213.  
THE TERM 128[S_1 ODD], OVER 73 DRAWS, GIVES A MULTIPLE OF 128, WHICH MODULO 256 IS EITHER 0 OR 128.  
IF WE IGNORE C_2 AND C_3 FOR A WHILE, THIS IMPLIES THAT WHEN AT SOME POSITION THE NUMBER IS IN [0,49] (THAT IS, | OR \),   
AFTER 73 DRAWS THE NUMBER IS IN [213,6] (MOD 256) OR [128,177].  
C_2 AND C_3 WILL STILL INCREASE THE NUMBER A BIT (ON AVERAGE BY ABOUT 12), BUT A VALUE IN [50,59] WON'T BE REACHED, WHICH IMPLIES NO EMPTY SPACE OCCURS BELOW | OR \. :-D

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DARKSTOORM                        2020-07-14 22:12
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THAT WAS A REALLY GREAT RESEARCH, arno! BD IS SO SIMPLE, YET IT HAS SO MUCH CONTENT TO TALK ABOUT!

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DUSTIN                            2020-07-14 22:46
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A VERY NICE AND CLEAR EXPLANATION OF THE SILENT CAVE!

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ARNO                              2020-07-15 07:37
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THX! I'VE ONE MORE RESEARCH QUESTION RELATED TO THE RNG TOPIC... SOMETHING FOR TONIGHT! :-D

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DUSTIN                            2020-07-15 15:54
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THE HORIZONTAL VS. VERTICAL STACK ISSUE? ;D

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ARNO                              2020-07-15 20:09
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YESS EXACTLY!   
I ALREADY HAVE THE ANSWER; JUST NEED TO WRITE A SMALL REPORT FOR THE FORUM! ;-D

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DUSTIN                            2020-07-16 22:19
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GREAT ARTICLE AGAIN! ;D

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ARNO                              2020-07-17 11:54
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THX! ;-)