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slime math 1: the wind maker
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by
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dustin
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DUSTIN                            2026-01-02 18:53
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Κ MUST CROSS THE STORMY SEA TO GET FROM HIS HOME ISLAND TO THE ONE WITH THE TREASURE - BUT HOW IS HE SUPPOSED TO MAKE IT THROUGH THE HAIL? IF ONLY HE COULD INFLUENCE THE WIND'S DIRECTION, HE MIGHT JUST BE ABLE TO SLIP UNDERNEATH THE FALLING STONES...

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HEYWOOD                           2026-01-02 19:01
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WOW, I HAVE NO IDEA HOW OR WHY THAT WORKS. :D

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DUSTIN                            2026-01-02 19:04
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HEE HEE, EXPLANATIONS WILL FOLLOW :D

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DUSTIN                            2026-01-02 22:18
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heywood, darkwood: I'M VERY SORRY FOR UPDATING THIS CAVE AND THEREBY DELETING YOUR SOLUTIONS! BUT, I FOUND A COMPLETELY UNINTENDED SOLUTION WHICH IS NOW FIXED. AND THIS IS ONE OF THE CAVES WHERE I DON'T WANT UNINTENDED SOLUTIONS!  
  
ASSUMING YOUR SOLUTIONS WERE AS INTENDED, THEY SHOULDN'T BE TOO HARD TO REPRODUCE!  
  
THANK YOU 4 PLAYING :)

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DARKSTOORM                        2026-01-02 23:57
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HAHAHA, I DESTROYED ONE SLIME (AND TWO LATER) IN THE RIGHT BOX AND AFTER A WHILE, LIKE 60 SECONDS, I NOTICED BOULDER SHAPES THAT MADE BIG "3":S, THEN "2":S :D  
  
SOMETIMES THERE ARE FUNNY PATTERNS LIKE DIAMOND SHAPES IN A GRID, OVALS OR STRAIGHT LINES OF BOULDERS :D

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DARKWOOD                          2026-01-03 09:19
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dustin, SEEMS TO BE SOLVABLE EVEN AFTER THE FIX  
I HOPE MY SOLUTION WAS INTENDED  
  
READING EXPLANATION,   
<a href="https://boulderdash.nl/forum/viewtopic.php?t=661" target="external">HTTPS://BOULDERDASH.NL/FORUM/VIEWTOPIC.PHP?T=661</a>  
BUT EVERY TIME THE CONVERSATION ABOUT SLIME PERMEABILITY COMES UP, IT'S ALWAYS NOT AS CLEAR AS THE FIRST TIME :)

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HEYWOOD                           2026-01-03 13:31
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DID THE UNINTENDED SOLUTION INVOLVE USING THE FIREFLIES TO DESTROY THE WALL AND THEN RUNNING THROUGH THE GAP FOR THE RETURN JOURNEY? I TRIED THAT A FEW TIMES BUT NEVER MANAGED IT!

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DARKWOOD                          2026-01-03 14:08
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heywood, YEAH, I ACCIDENTALLY RELEASED A FIREFLY AND IT DESTROYED PART OF THE M-WALL, BUT THERE ARE STILL TOO MANY BOULDERS TO GET THROUGH.

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DUSTIN                            2026-01-03 14:22
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ALRIGHT GUYS, SO HERE'S THE EXPLANATION HOW THIS WORKS :D  
  
darkwood:   
Α YES, YOUR SOLUTIONS ARE INTENDED!  
Α THE UNINTENDED SOLUTION WENT COMPLETELY DIFFERENT, THE NEW VOODOOS PREVENT THAT ONE.   
Α INDEED, THE EXPLANATIONS YOU READ IN THE FORUM ARE SIMILAR BUT NOT QUITE THE SAME :D THIS CAVE IS NOT ABOUT IMPERMEABILITY, BUT IT'S ANOTHER EFFECT, ALTHOUGH THE MATH BEHIND IT IS SIMILAR!  
  
heywood: I THINK IT'S A GOOD OPPORTUNITY TO MAKE ANOTHER EPISODE OF "JUST ONE!" SERIES, SO I DON'T WANNA SPOIL IT YET :D  
  
EXPLANATION: THE SLIME FORMULA  
------------------------------  
THE SLIME FORMULA GOES AS FOLLOWS:  
  
F * S = 949 - D  
  
F = FRAME DIFFERENCE  
S = TOTAL SLIME COUNT IN THE CAVE  
949 = PERIODIC LENGTH OF THE RNG OF THE BD ENGINE  
D = DISTANCE BETWEEN TWO SLIMES (NOT GEOMETRICALLY, IT ONLY COUNTS HOW MANY OTHER SLIMES ARE IN-BETWEEN.)  
  
THE FORMULA WORKS AS FOLLOWS:  
"WHENEVER A SPECIFIC PIECE OF Ι IS (IN)ACTIVE IN A SPECIFIC FRAME, THEN THE D-TH NEXT Ι WILL ALSO BE (IN)ACTIVE IN THE F-TH NEXT FRAME."  
  
WHY DOES THIS FORMULA WORK:  
--------------------------  
Β THE RNG HAS A PERIOD OF 949, SO AFTER 949 SCANS, THE RNG REPEATS THE SAME INPUT SEQUENCE.  
  
Β WHENEVER A SLIME IS SCANNED, THE RNG GENERATES A NEW INPUT, DECIDING IF THE SLIME IS ACTIVE IN THIS FRAME OR NOT.  
  
Β THIS HAPPENS REGARDLESS WHETHER THE SLIME ACTUALLY HAS SOMETHING TO SWALLOW! (IF IT HASN'T, THE INPUT IS IRRELEVANT, BUT IT'S STILL GENERATED!)  
  
| SO EVERY 949TH SLIME IN THE SCANNER - FRAME BY FRAME - GETS THE SAME INPUT.  
  
| EXAMPLE S=315, F=3, D=4: WE HAVE 315 SLIMES IN A CAVE. IN FRAME 1, SLIME 1 GETS INPUT 1, SLIME 2 GETS INPUT 2, AND SO ON..., SLIME 315 GETS INPUT 315. IN FRAME 2, SLIME 1 GETS INPUT 316, SLIME 2 GETS INPUT 317... IN FRAME 3, SLIME 1 GETS INPUT 631... IN FRAME 4, SLIME 1 GETS INPUT 946, SLIME 4 GETS INPUT 949, AND THEN SLIME 5 GETS INPUT 1 AGAIN.   
SO WE HAVE FRAME 1, SLIME 1 = SAME INPUT AS FRAME 4, SLIME 5! (3 FRAMES LATER, THE 4TH NEXT SLIME)  
  
HOW DOES THE FORMULA WORK IN THIS CAVE:  
--------------------------------------  
  
| AT THE BEGINNING, THERE ARE 478 SLIMES IN THE CAVE, SO S=478.  
| THE FORMULA WORKS WITH F=2, D=-7.  
| THAT MEANS THE FALLING ΒΒΒ HAVE A "7 LEFT, 2 UP" PATTERN AT THE BEGINNING (WHICH IS NOT SO CLEARLY VISIBLE AND ALSO NOT HELPFUL FOR Κ TO MAKE IT THROUGH THE HAILS).  
  
| IN THE SOLUTION, 3 SLIMES ARE DESTROYED AT THE RIGHT SIDE. THIS CHANGES S TO S=475.  
| NOW THE FORMULA WORKS WITH F=2 AND D=-1.  
| THAT GIVES A ΒΒΒ PATTERN OF "1 LEFT, 2 UP", WHICH IS CLEARLY VISIBLE AND WHICH Κ CAN PASS THROUGH TO GET TO THE LEFT SIDE!  
  
| ON THE LEFT ISLAND, Κ DESTROYS EXACTLY ONE MORE Ι, SO NOW S=474.  
| THE FORMULA NOW WORKS WITH F=2, D=+1.  
| SO THE PATTERN CHANGES TO "1 RIGHT, 2 UP", ALLOWING Κ TO RETURN TO THE STARTING AREA! :D  
  
WHY DON'T SUCH CLEAR PATTERNS OCCUR IN NORMAL, CLASSIC SLIME CAVES:  
------------------------------------------------------------------  
  
| OF COURSE, THE FORMULA ALWAYS WORKS FOR A GIVEN SLIME COUNT S WITH SOME VALUES FOR F AND D, SO SLIME PATTERNS DO, TECHNICALLY, OCCUR IN EVERY SLIME CAVE!   
  
| HOWEVER, FOR THE PATTERNS TO BE THAT EVIDENT AS IN THIS CAVE, F AND D NEED TO BE RELATIVELY SMALL NUMBERS.  
  
| THAT MEANS THE FORMULA CAN ONLY WORK IF S IS UNNATURALLY HIGH, HIGHER THAN IN ANY NORMAL CAVE.  
  
EXAMPLE: BOULDER DASH 2 CAVE E  
------------------------------  
  
| TOTAL SLIME COUNT: S=4*19=76  
  
| THE SMALLEST POSSIBLE VALUES FOR F AND D TO MAKE THE FORMULA WORK ARE: F=12, D=37.  
  
| THESE VALUES ARE MUCH TOO HIGH TO PRODUCE ANY VISIBLE PATTERN!  
  
SPECIAL CASE: IMPERMEABILITY (N O T IN THIS CAVE, BUT IN GENERAL):  
-----------------------------------------------------------------  
  
| THE WHOLE "SLIME IMPERMEABILITY" THING IS ALSO INCLUDED IN THE FORMULA, NAMELY FOR D=0, SO F*S=949.  
  
| THE FORMULA NOW SAYS: "WHENEVER A SPECIFIC SLIME IS (IN)ACTIVE IN A SPECIFIC FRAME, THE SAME SLIME WILL ALSO BE (IN)ACTIVE F FRAMES LATER."  
  
| IN OTHER WORDS, THE PATTERN OF EACH SPECIFIC SLIME REPEATS AFTER F FRAMES.  
  
| AS THE PRIMA FACTORISATION OF 949 IS 13*73, THERE ARE ONLY A FEW POSSIBILITIES TO MAKE THE FORMULA WORK:  
  
| CASE 1: F=949, S=1 (BORING CASE)  
IF THERE'S ONLY ONE Ι IN THE CAVE, ITS PATTERN WILL REPEAT AFTER 949 FRAMES. WITH NORMAL CAVE SPEED, THIS IS MORE THAN TEN MINUTES, SO IT'LL NEVER BE NOTICED!  
  
| CASE 2: F=73, S=13 (IMPERMEABILITY WITH 13 SLIMES)  
Α WITH 13 SLIMES IN THE CAVE, THERE HAVE TO BE 73 "INACTIVE" RNG INPUTS IN A ROW TO MAKE A SLIME IMPERMEABLE. THIS IS RELATIVELY UNLIKELY TO HAPPEN, SO IMPERMEABILITY WITH 13 SLIMES ONLY OCCURS WITH PERMEABILITY 4 OR HIGHER.  
  
| CASE 3: F=13, S=73  
WITH 73 SLIMES IN THE CAVE, THERE ARE ONLY 13 INDEPENDANT RNG INPUTS FOR EACH SLIME, SO IMPERMEABILITY BECOMES MORE LIKELY. CONSEQUENTLY, IMPERMEABILITY WITH 73 SLIMES OCCURS ALREADY WITH PERMEABILITY 3 OR HIGHER.  
  
| CASE 4: F=1, S=949  
THE ULTIMATE EXTREME CASE - WITH 949 SLIMES IN A CAVE, EVERY SLIME GETS ALWAYS THE SAME INPUT FROM THE RNG, SO EACH SLIME IS EITHER IMPERMEABLE OR 100 PERCENT PERMEABLE! :D

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HEYWOOD                           2026-01-03 15:42
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MIND BLOWN

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ARNO                              2026-01-03 17:34
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GREAT WORK AND VERY FASCINATING! :-D  
  
NOW THAT THIS SERIES OF CAVES HAS STARTED, I HAVE ALSO REFRESHED THE FORUM THREAD MENTIONED BELOW WITH AN OVERVIEW OF PREVIOUS "SLIME PATTERN" CAVES, EACH WITH A DESCRIPTION HOW THEY WORK.  
(HOPE I HAVEN'T FORGOTTEN ANY.)  
SEE:  
<a href="https://www.boulder-dash.nl/forum/viewtopic.php?t=661" target="external">HTTPS://WWW.BOULDER-DASH.NL/FORUM/VIEWTOPIC.PHP?T=661</a>

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DUSTIN                            2026-01-04 18:35
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EXCELLENT IDEA TO REVIVE THAT TOPIC arno :D  
  
I JUST RE-READ IT AND ALSO READ YOUR NEW COMMENTS ON THE CAVES - VERY GOOD AND TO THE POINT IMO!  
  
THERE'S SURELY GONNA BE MORE STUFF TO DISCUSS THERE SOON ;)