Talk:Firework Star/Archive 1

Crafting
So, uh... is the crafting square messed up, or do we just not know how to craft these things yet? Cobalt32 18:35, 7 December 2012 (UTC)


 * I personally do not know how to craft it, and I set the page up, if you do, I can put it in. Cultist O 18:36, 7 December 2012 (UTC)


 * Ah. I didn't know how to craft it either; I was just wondering why the grid was blank. Cobalt32 18:38, 7 December 2012 (UTC)


 * K, well it's a shapeless recipe of gunpowder, dye, and special effect modifiers (SEM optional) not sure how many SEMs you can have though, or how to depict that. Also you can have any number of dyes, so long as there is at least one, much like dying armour. Cultist O 18:47, 7 December 2012 (UTC)

What's the data value?
Data value is missing... Cultist O 19:29, 7 December 2012 (UTC)
 * Just did some experimenting. It's 402, right after firework rocket (401). --Kahless61 22:19, 7 December 2012 (UTC)

Recipe
The recipe visual is a little confusing. You only need one dye and a gunpowder to make it, but the recipe could make someone glance at it and think you need 8 dyes. --Kahless61 19:41, 7 December 2012 (UTC)


 * I agree, there must be a better way of displaying that, though the armour page doesn't bother to show dying leather armour. Cultist O 19:47, 7 December 2012 (UTC)


 * Also isn't the star shapeless to create and fade? Cultist O 20:10, 7 December 2012 (UTC)


 * Fixed --Kahless61 20:19, 7 December 2012 (UTC)

"Multiple Firework Stars can be crafted with Firework Rocket for multiple types of explosion." This statement is unclear. I believe it's talking about how you can put multiple stars into the rocket recipe (you cannot craft them WITH the rocket, however). The result will be all of the stars effects going off when the rocket explodes (I guess that could be interpreted as "multiple types of explosion" but I'd probably phrase it some other way). Anyone want to clean this up? I'm not very good at phrasing things. --Kahless61 21:09, 7 December 2012 (UTC)


 * Having not actually tested anything I'm sort of going off what it seems like they meant, but how's that for wording? Cultist O 21:41, 7 December 2012 (UTC)


 * Perfect. I've been doing testing. That's exactly the way it works. Thanks. --Kahless61 21:43, 7 December 2012 (UTC)

Diamonds??
That had better be some trail... --Mental Mouse 03:13, 8 December 2012 (UTC)

Possible Combinations
Not sure if anyone is interested in how many possible combinations there are for making fireworks, but being a math junky, I decided to figure it out, and although there were probably some redundancies in my math, I'm quite sure I'm correct or at least very close.

My calculations take into account:

On Firework Stars: leaving blank slots, all 16 colors, all 5 shapes, and the 2 effects.

On Firework Rockets: all the above with the addition of the 3 flight times.

Possible Firework Stars: 2,035,138,560 (2.0 * 109, 2 billion)

Possible Firework Rockets: 144,596,532,472,853,875,918,507,017,587,300, 000,000,000,000,000,000,000,000,000,000,000 (1.4 * 1065, 145 vigintillion) 199.187.203.27 00:36, 4 February 2013 (UTC)


 * Can you explain your math? Not all of us have the motivation to do so ourselves, and there are plenty who may be interested but wouldn't actually know how to do so. ;) 「 ディノ 奴 千？！ 」? · ☎ Dinoguy1000 17:47, 4 February 2013 (UTC)


 * Okay, so it's been about 2 weeks since I did the math so bare with me on recalling all the numbers as there were a lot. I will try to display the numbers in a box format seeing that the possibilities are based upon the 3 x 3 crafting grid.


 * First I would like to state how this is done. It is basically done using factorials. I.E. 3! (3 factorial) = 3 * 2 * 1 = 6. This is used to represent using up one of the options.


 * With that being said, there could be more rocket combinations than I stated I just didn't feel the need to put in the redundancy of using the same Firework Star multiple times.


 * C = Color
 * G = Gun Powder
 * S = Shape
 * E = Effect
 * Firework Star:
 * Possibilities include, blank slot, 16 colors, 5 shapes, and 2 effects.


 * The format I'll be following is as such:


 * C C C
 * C G C
 * E S E


 * This is for the reason that it's symmetrical. Now another thing, where I have the Shape and Effect slots at, the numbers will be bigger for the fact that it's possible to use a color instead. With all this in mind, here comes the numbers.


 * 16 16 15
 * 14 01 13
 * 14 16 13


 * When you multiply all the numbers together you should get something close to my number, Again, it's been a while so I may have missed something.


 * Firework Rocket:


 * For simplicity sake, I'm gonna call whatever the above comes out as, F (Firework Star Possibilities).


 * Format is as follows:


 * F = Firework Star
 * P = Paper
 * G = Gunpowder


 * F F F
 * F P F
 * G G G


 * The math for this should look like:


 * F+1 F+0 F-1
 * F-2 001 F-3
 * F-3 001 F-4


 * This is how I arrived at my numbers, and should work out roughly the same for everyone else. the "1"s in the equations represent the fact that the material has to exist (i.e. You can't make a Rocket without at least 1 paper and 1 gunpowder). Also another note, they always start out as 1 more than the number you're using because the possibility of leaving that slot blank. Hope this helps and that someone at least finds it interesting or useful.


 * Good day to all!
 * 199.187.203.27 20:32, 17 February 2013 (UTC)

--

I found something completely different, considering every different possible description for a Firework Star and for a Firework Rocket. I hope I will make myself clear, and I may be absolutely or slightly wrong, but here is my reasoning, please correct me if I'm wrong.

Firework Star
The order of the dyes on the crafting table matters, as it is kept in the Firework Star description. The order of the shapes and effects doesn't matter. With the addition of effects and shape, we've got:
 * 16 * 20 possibilities with 1 dye: there are 16 colors, and we can combine the color we use with ((1 * nothing) + (4 * shape) + (2 * one effect) + (1 * two effects) + (8 * (shape and effect)) + (4 * (shape and two effects))).

To those 16 * 20 possibilities we add:
 * (16 ^ 2) * 20 possibilities with two dyes: there are 16 * 16 combinations of two colors. Each of those combinations can be combined with the 20 combinations of shapes and effects listed above.
 * The same applies with 3, 4 and 5 dyes, adding ((16 ^ 3) + (16 ^ 4) + (16 ^ 5)) * 20 possibilities to the total.

The total is now (16 + (16 ^ 2) + (16 ^ 3) + (16 ^ 4) + (16 ^ 5)) * 20 possibilities.
 * With 6 dyes, there is room for only two shapes and effects, adding (16 ^ 6) * 16, as we can combine each color combination with ((1 * nothing) + (4 * shape) + (2 * one effect) + (1 * two effects) + (8 * (shape and effect))).
 * With 7 dyes, there is room for only one shape or effect, adding (16 ^ 7) * 7 possibilities, as we can combine each color combination with ((1 * nothing) + (4 * shape) + (2 * effect)).
 * With 8 dyes, there is no room for any shape or effect, adding (16 ^ 8) possibilities.

The total is (16 + (16 ^ 2) + (16 ^ 3) + (16 ^ 4) + (16 ^ 5)) * 20 + (16 ^ 6) * 16 + (16 ^ 7) * 7 + (16 ^ 8) = 6 464 820 544 "basic" Firework Stars. With the addition of the fading effect, it is possible to combine a Firework Star with zero to height dyes. The color order matters as it changes the item description. This gives us 1 + 16 + (16 ^ 2) + (16 ^ 3) + (16 ^ 4) + (16 ^ 5) + (16 ^ 6) + (16 ^ 7) + (16 ^ 8) = 4 581 298 449 possible combinations for each different Firework Star, for a grand total of 6 464 820 544 * 4 581 298 449 = 29 617 272 331 290 537 984 (almost 3 * 10 ^ 19) different Firework Stars. Let's call this number F.

Firework Rocket
The order of the Firework Stars matters, as it changes the description of the resulting Firework Rocket. When we craft a Firework Rocket, we can use 0 to 8 Firework Stars:
 * With no Firework Star, there is 1 possible Firework Rocket.
 * With 1 Firework Star, there are F different Firework Rockets.
 * With 2 Firework Stars, there are F ^ 2 different Firework Rockets...

And so on, for a total of 1 + F + F ^ 2 + F ^ 3 + F ^ 4 + F ^ 5 + F ^ 6 + F ^ 7 + F ^ 8 = 1 + 29617272331290537984 + (29617272331290537984 ^ 2) + (29617272331290537984 ^ 3) + (29617272331290537984 ^ 4) + (29617272331290537984 ^ 5) + (29617272331290537984 ^ 6) + (29617272331290537984 ^ 7) + (29617272331290537984 ^ 8) = 1 + 29617272331290537984 + 877182820345828058750339853728146784256 + 25979762474511892143932594053669186249668885871489821179904 + 769449700309861263739094623922315132647847384609171621348337248158353809473536 + 22789001319307050497706012074971683525653312722148132070854539043513650253661039278095577250791424 + 674948058232056273299612861129000441465762317372797488626668341180447861609517974285024121454540594261050546733449216 + 19990120450134555088575054298746712745363599118346368260681572208743201478318885327968154767483683366362883120663775492800476616583020544 + 592052841306935312947348476063265161079110776350040932596249455027768587883650895750009679296435280362889052473179577731220727873443647112636939821484343296 = 592 052 841 306 935 312 967 338 596 513 399 716 168 360 778 707 019 701 614 935 456 008 563 263 636 296 705 737 900 959 808 170 252 419 146 200 061 368 426 268 073 631 952 451 281 025 067 718 465 119 580 161 different Firework Rockets. That's quite a lot! (almost 6 * 10 ^ 155 or 6 * 10 ^ 55 googol) Again, I may be wrong. If you think I made an error, please point it out! 82.226.69.1 07:20, 7 February 2014 (UTC) Jidehem1993


 * You wrote "height" instead of "eight".
 * You miscalculated the number of firework stars, getting the last 4 digits wrong.
 * You forgot that a rocket must have 1 to 3 gunpowder for fuel, and therefore can't have 8 firework stars.

Permutations and Combinations
To correctly calculate the number of different firework stars, it is necessary to handle the effect ingredients (diamond and glowstone dust) specially, because you can include effects only once each, whereas the same color can be used multiple times. Also, the order of the colors in the crafting grid determines their order in the description, whereas the positioning of an effect ingredient doesn't matter, because effects are always listed after the colors, and Trail is always listed before Twinkle. To make a firework star, you must use 1 gunpowder, 1 to 8 units of dye, 0 or 1 shape ingredients (there are 4 different shape ingredients), 0 or 1 diamonds, and 0 or 1 glowstone dust. Of course, there can only be 9 ingredients total since that is the number of slots in the crafting grid. (You can actually use multiple diamonds and/or multiple glowstone dust, but that does not give a unique result and therefore the ability to use redundant effect ingredients does not affect the number of unique firework stars.) There are 5*2*2=20 possibilities for shape and effects. 1 possibility uses 0 ingredients for shape and effects, 6 possibilities use 1 ingredient for shape or effect, 2*4+1=9 possibilities use 2 ingredients, and 4 possibilities use 3 ingredients. Depending on how many crafting slots are used up by dye, you could have 1+6+9+4=20, 1+6+9=16, 1+6=7, or 1 possibility available. The hotkey for the exponentiation (x to the power of y) key in Calc is 'y', so with that in mind: 16*20+16y2*20+16y3*20+16y4*20+16y5*20+16y6*16+16y7*7+16y8*1=6464820544 6,464,820,544 firework stars with no fade Fades don't use shape or effect ingredients; the shape and effects initially included in the firework star carry over to the fade. So the number of fades is simply: 16+16y2+16y3+16y4+16y5+16y6+16y7+16y8=4581298448 4,581,298,448 fades 6464820544*(4581298448+1)=29617272331290536256 (must add 1 to represent the lack of a fade) 29,617,272,331,290,536,256 firework stars

A firework rocket must be made with 1 paper, 1 to 3 gunpowder, and 0 to 7 firework stars. The firework stars are listed in the description according to the order of their positions in the crafting grid. The flight duration (which is equal to the amount of gunpowder used) is listed before the firework stars. With a maximum of 9 total ingredients, a 7-star rocket can have only 1 gunpowder, and a 6-star rocket can have only 1 or 2 gunpowder. So: 1*3+29617272331290536256*3+29617272331290536256y2*3+29617272331290536256y3*3+29617272331290536256y4*3+29617272331290536256y5*3+29617272331290536256y6*2+29617272331290536256y7*1= 1.999012045013454692575323804026e+136 firework rockets Or, using the bignum calculator at https://defuse.ca/big-number-calculator.htm: 1*3+29617272331290536256*3+29617272331290536256^2*3+29617272331290536256^3*3+29617272331290536256^4*3+29617272331290536256^5*3+29617272331290536256^6*2+29617272331290536256^7*1 19,990,120,450,134,546,925,753,238,040,259,571,986,820,916,295,504,403,597,177,411,829,216,667,290,445,679,938,450,910,650,975,366,847,498,729,147,381,808,461,242,200,332,347,927,491 firework rockets

However, there are probably more players interested in the number of visually/functionally distinct firework rockets than in the number of different descriptions. To calculate this, one must know exactly how the fireworks work. A rocket has a random starting velocity and a flight duration dependent on the amount of gunpowder used in the final crafting step. When the rocket runs out of fuel (always when it runs out of fuel, not upon hitting something), all of its firework stars explode simultaneously. The explosions will be centered on the same location, but all rotated to different random directions on a vertical axis. Having multiple identical firework stars is distinct from having only 1, because more firework stars means more explosion particles. The particles in a shape are NOT randomly arranged within it, but are randomly colored. Each particle starts with a color randomly chosen from the list of starting colors, and if there is a fade, each particle fades to a color randomly chosen from the list of fade colors. A particle's fade color is chosen independently of its starting color, and because of this, a firework star with a fade color list that totally matches its starting color list is distinct from a star with the same starting color list but no fade, UNLESS only one color is used. For example, if a firework star's starting color list consists of red 1 or more times, and the fade color list also consists of red 1 or more times, you won't be able to see a fade happening. With 16 colors, 5 shapes, Trail, and Twinkle, there are 16*5*2*2=320 firework stars that are distinct from each other and that have pointless fades. So 320 must be subtracted from the count of firework stars.

The order of colors in a firework star doesn't matter, and even the total amount of dye doesn't matter. Only the ratio matters. So 4 red dye + 2 blue dye is equivalent to 2 red dye + 1 blue dye. And 2 green dye + 1 yellow dye is distinct from 2 yellow dye + 1 green dye. But when the mix consists of equal amounts of the colors, the order doesn't matter.

So it seems necessary to tediously list all possible ratios. Care must be taken to avoid ratios that are equal to previously listed ratios. (The redundant ratios are 2, 3, 4, 5, 6, 7, 8, 2/2, 3/3, 4/4, 2/2/2, 4/2, 2/2/2/2, 4/2/2, and 6/2.)

Because using 6, 7, or 8 units of dye leaves little or no room for shape and effect ingredients, ratios requiring those amounts of dye must be listed on separate lists.

When a ratio doesn't contain the same number more than once, the permutations for that ratio are used as-is. When a ratio consists of just the same number multiple times, the permutations must be divided by an appropriate factorial to get combinations instead of permutations. When a ratio has 2 or more different numbers, at least one of which is repeated, then there's kind of a weird hybrid permutation/combination thing.

ratio		number of permutations/combinations 1		16=				16 1/1		16*15/2=			120 1/1/1		16*15*14/3!=			560 2/1		16*15=				240 1/1/1/1		16*15*14*13/4!=			1820 2/1/1		16*15*14/2=			1680 3/1		16*15=				240 1/1/1/1/1	16*15*14*13*12/5!=		4368 2/1/1/1		16*15*14*13/3!=			7280 2/2/1		16*15*14/2=			1680 3/1/1		16*15*14/2=			1680 3/2		16*15=				240 4/1		16*15=				240 16+120+560+240+1820+1680+240+4368+7280+1680+1680+240+240=20164 20,164 total color mixes that need 5 dye or less

1/1/1/1/1/1	16*15*14*13*12*11/6!=		8008 2/1/1/1/1	16*15*14*13*12/4!=		21840 2/2/1/1		16*15*14*13/2/2=		10920 3/1/1/1		16*15*14*13/3!=			7280 3/2/1		16*15*14=			3360 4/1/1		16*15*14/2=			1680 5/1		16*15=				240 8008+21840+10920+7280+3360+1680+240=53328 53,328 total color mixes that need 6 dye

1/1/1/1/1/1/1	16*15*14*13*12*11*10/7!=	11440 2/1/1/1/1/1	16*15*14*13*12*11/5!=		48048 2/2/1/1/1	16*15*14*13*12/2/3!=		43680 2/2/2/1		16*15*14*13/3!=			7280 3/1/1/1/1	16*15*14*13*12/4!=		21840 3/2/1/1		16*15*14*13/2=			21840 3/2/2		16*15*14/2=			1680 3/3/1		16*15*14/2=			1680 4/1/1/1		16*15*14*13/3!=			7280 4/2/1		16*15*14=			3360 4/3		16*15=				240 5/2		16*15=				240 6/1		16*15=				240 11440+48048+43680+7280+21840+21840+1680+1680+7280+3360+240+240+240=168848 168,848 total color mixes that need 7 dye

1/1/1/1/1/1/1/1	16*15*14*13*12*11*10*9/8!=	12870 2/1/1/1/1/1/1	16*15*14*13*12*11*10/6!=	80080 2/2/1/1/1/1	16*15*14*13*12*11/2/4!=		120120 2/2/2/1/1	16*15*14*13*12/3!/2=		43680 3/1/1/1/1/1	16*15*14*13*12*11/5!=		48048 3/2/1/1/1	16*15*14*13*12/3!=		87360 3/2/2/1		16*15*14*13/2=			21840 3/3/1/1		16*15*14*13/2/2=		10920 3/3/2		16*15*14/2=			1680 4/1/1/1/1	16*15*14*13*12/4!=		21840 4/2/1/1		16*15*14*13/2=			21840 4/3/1		16*15*14=			3360 5/1/1/1		16*15*14*13/3!=			7280 5/2/1		16*15*14=			3360 5/3		16*15=				240 6/1/1		16*15*14/2=			1680 7/1		16*15=				240 12870+80080+120120+43680+48048+87360+21840+10920+1680+21840+21840+3360+7280+3360+240+1680+240=486438 486,438 total color mixes that need 8 dye

20,164 total color mixes that need 5 dye or less 53,328 total color mixes that need 6 dye 168,848 total color mixes that need 7 dye 486,438 total color mixes that need 8 dye (Now to include shapes and effects: 1 possibility uses 0 ingredients, 6 possibilities use 1, 9 possibilities use 2, and 4 possibilities use 3. 20, 16, 7, 1) 20164*20+53328*16+168848*7+486438=2924902 2,924,902 visually/functionally distinct firework stars without fades 20164+53328+168848+486438=728778 728,778 visually/functionally distinct fades (320 must be subtracted from the count of firework stars, because there are 320 with pointless fades) 2924902*(728778+1)-320=2131607154338 2,131,607,154,338 visually/functionally distinct firework stars

1+2131607154338*3+2131607154338y2/2*3+2131607154338y3/3!*3+2131607154338y4/4!*3+2131607154338y5/5!*3+2131607154338y6/6!*2+2131607154338y7/7!= 3.9675247803502678013047485962163e+82 visually/functionally distinct firework rockets

1+2131607154338*3+2131607154338^2/2*3+2131607154338^3/(1*2*3)*3+2131607154338^4/(1*2*3*4)*3+2131607154338^5/(1*2*3*4*5)*3+2131607154338^6/(1*2*3*4*5*6)*2+2131607154338^7/(1*2*3*4*5*6*7) 39,675,247,803,502,678,013,047,485,962,163,290,620,857,699,449,061,223,588,692,411,165,364,365,964,747,258,227 visually/functionally distinct firework rockets

The blank firework star available only in Creative Mode is not included in any of this. But if you really must know how many different firework rockets can be made in Creative Mode, add 3 to those numbers. Adding additional blank stars to a blank star rocket has no effect, and adding blank stars to a rocket with non-blank stars has no effect. So there are only 3 different blank star rockets, distinguished only by their flight duration. They are distinct (both in description and function) from starless rockets, because starless rockets have no flight time listed, and seem to have a flight duration of 0.5.

SUMMARY: 2.962e19 stars and 1.999e136 rockets, but only 2.132e12 distinct stars, and only 3.968e82 distinct rockets

not 2.035e9 stars not 1.446e65 rockets not 5.921e155 rockets