Recently I have been trying to devise a simple calculation to figure out exactly how many different banner combinations there are, so I went to the wiki just to see if the answer was already well-known. What the wiki says is this:

With 16 blank flags, 33 patterns of 16 colors each (528 uniquely-colored patterns), and 0 to 6 patterns per flag, the number of uniquely crafted banners is 16 × ( 528^{0} + 528^{1} + 528^{2} + 528^{3} + 528^{4} + 528^{5} + 528^{6} ) ≈ 347 quadrillion. The number of visually distinct flags is smaller, because patterns may completely cover other patterns.

Correct me if I'm wrong (which is highly probable to be honest), but this calculation seems to leave out the fact that the order in which patterns are added to the banner results in more unique patterns. For example, there is a limit of 6 patterns you can add (in vanilla), but for any of those 6 different patterns you add, there are six different "slots" in the order in which you can add them to. So with these 6 patterns there are 36 different combinations that stem from the fact that the order in which patterns are added cause them to overlap in different ways.

So if this is correct the actual calculation should be:

(Carrots denote powers raised to another power because superscripts cannot be added onto other superscripts.)

16 x (528^{(}^{0}^{^2)} + 528^{(1^2)} + 528^{(2^2)} + 528^{(3^2)} + 528^{(4^2)} + 528^{(5^2)} + 528^{(6^2)}) = 1.6555x10^{99} different combinations

The powers are squared because for every "n" number of patterns there are added, there are "n" number of "slots" in the crafting order each of those patterns could be placed to make another unique banner. Simplified it is:

16 x (528^{0} + 528^{1} + 528^{4} + 528^{9} + 528^{16} + 528^{25} + 528^{36}) = 1.6555x10^{99} different combinations

However, this does result in many more banners that are not visually distinct. This is because when adding patterns of the same color, order does not matter one bit and the banners will look identical except when the cursor is hovered over the item in the inventory to view the list of patterns. The order does however, still matter when patterns of different colors are added.

Please discuss and correct me if I missed something that would make this entire calculation bogus.

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Sorry to tell you, but the old numbers are correct.

For example using 3 layers: the first layer has 528 possible patterns, the second layer has 528 possible patterns and the third layer has 528 possible patterns.

You get all possible combinations of patterns by multiplying.

The order of layers is already built into the calculation. If the order were unimportant you would have to divide the terms by 1, 1, 2, 6 ,24, 120. 720 respectively, which is n! (n Factorial), which you get by multiplying all numbers from 1 to n.

Contrary to your 36 possible combinations there's actually 720 different combinations for 6 layers. You get to this number like so:

In first place can be any of the 6 layers, in second place there's 5 possible layers left, third place chooses from 4 layers and so on. This means you have 6 x 5 x 4 x 3 x 2 x 1 combinations (which is 720 or 6 Factorial).

With the latest snapshot, 14w31a, it appears they've added 5 more patterns (the reverse of 5 existing patterns) which technically makes 38 patterns instead of 33. That means there are now 608 pattern-color combinations and a correspondingly higher total possible combinations of pattern/colors. Someone else can do the math on that one but I know the number is huge.

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I went ahead and did the math, it's 110,447,470,856,065,619,857,908,957,184 different banners for 10 layers. I just picked 10 because it sounded like a reasonable number, and I haven't found many unique banners with more than 10 layers.

With 16 blank banners, 38 patterns of 16 colors each (608 uniquely-colored patterns), and 0 to 6 patterns per banner, the number of uniquely crafted banners is 16 × ( 608^{0} + 608^{1} + 608^{2} + 608^{3} + 608^{4} + 608^{5} + 608^{6}) ≈ 809 quadrillion (809,573,616,779,945,488). The number of visually distinct flags is smaller, because one or more patterns may completely cover other patterns, or the entire banner, or be duplicated due to the symmetric set of patterns (e.g. field Or (yellow) + per pale azure (blue) = field azure + per pale Or inverted).

So, you were off by about ≈ 450 quadrillion.

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^{0}+ 528^{1}+ 528^{2}+ 528^{3}+ 528^{4}+ 528^{5}+ 528^{6}) ≈ 347 quadrillion. The number of visually distinct flags is smaller, because patterns may completely cover other patterns.So if this is correct the actual calculation should be:

(Carrots denote powers raised to another power because superscripts cannot be added onto other superscripts.)

16 x (528

^{(}^{0}^{^2)}+ 528^{(1^2)}+ 528^{(2^2)}+ 528^{(3^2)}+ 528^{(4^2)}+ 528^{(5^2)}+ 528^{(6^2)}) = 1.6555x10^{99}different combinationsThe powers are squared because for every "n" number of patterns there are added, there are "n" number of "slots" in the crafting order each of those patterns could be placed to make another unique banner. Simplified it is:

16 x (528

^{0}+ 528^{1}+ 528^{4}+ 528^{9}+ 528^{16}+ 528^{25}+ 528^{36}) = 1.6555x10^{99}different combinationsHowever, this does result in many more banners that are not visually distinct. This is because when adding patterns of the same color, order does not matter one bit and the banners will look identical except when the cursor is hovered over the item in the inventory to view the list of patterns. The order does however, still matter when patterns of different colors are added.

Please discuss and correct me if I missed something that would make this entire calculation bogus.

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You think magic isn't real? Consider this: for every person, there is a sentence -- a series of words -- which has the power to destroy them.

For example using 3 layers: the first layer has 528 possible patterns, the second layer has 528 possible patterns and the third layer has 528 possible patterns.

You get all possible combinations of patterns by multiplying.

The order of layers is already built into the calculation. If the order were unimportant you would have to divide the terms by 1, 1, 2, 6 ,24, 120. 720 respectively, which is n! (n Factorial), which you get by multiplying all numbers from 1 to n.

Contrary to your 36 possible combinations there's actually 720 different combinations for 6 layers. You get to this number like so:

In first place can be any of the 6 layers, in second place there's 5 possible layers left, third place chooses from 4 layers and so on. This means you have 6 x 5 x 4 x 3 x 2 x 1 combinations (which is 720 or 6 Factorial).

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Co-Owner of Verlax NetworkFrom the Minecraft Wiki:^{0}+ 608^{1}+ 608^{2}+ 608^{3}+ 608^{4}+ 608^{5}+ 608^{6}) ≈ 809 quadrillion (809,573,616,779,945,488). The number of visually distinct flags is smaller, because one or more patterns may completely cover other patterns, or the entire banner, or be duplicated due to the symmetric set of patterns (e.g. field Or (yellow) + per pale azure (blue) = field azure + per pale Or inverted).If you're reading this, then you're definitely not a sandwich.

Sorry, my dear, but I don't give a snickerdoodle.

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