Kaleidograph Pattern Combinations

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How Many Different
Combinations Are There?

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With nine die-cut cards and three backgrounds, each having two sides with different colors, we’ve long wondered —
how many different combinations does every Kaleidograph make?

Let’s say a unique pattern consists of at least one die-cut card and one background color. Obviously the die-cut cards can be stacked and flipped in different orders and numbers but the Kaleidograph is also a paper kaleidoscope and its patterns change when the cards are rotated 90º, or a quarter turn.

We asked our technically-minded friends — mathematicians, architects, physicists,
for help in figuring out simply how to frame the problem but to no avail.
Everyone recognized it as some kind of factorial question but how was it expressed?

 

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Here’s the answer:

Each die-cut card can be used in four ways, that is, in two rotations and in two colors/sides.
Therefore the formula for calculating how many combinations of the nine pierced cards would be:

(9×4)x(8×4)x(7×4)x(6×4)x(5×4)x(4×4)x(3×4)x(2×4)x(1×4) or 36x32x28x24x20x16x12x8x4= 95,126,814,720!

 

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But since there are six different background colors, the final mind-Boggling answer is 95,126,814,720 x 6 = 570,760,888,320!

 

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That is, over 570 billion combinations within just one set of Kaleidograph … nearly infinite if you combine sets!

So, if you create one Kaleidograph pattern a second it would take you over 18,000 years to make them all.

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