How many different combinations are there?

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? 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:

(9x4)x(8x4)x(7x4)x(6x4)x(5x4)x(4x4)x(3x4)x(2x4)x(1x4) or 36x32x28x24x20x16x12x8x4= 95,126,814,720!

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

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.