Problems with every day objects (repost): Baby Building Blocks

Discrete Mathematics Level 5





Cube Net depicted above

Today, while sitting in the doctors office waiting, I saw four identical cubes with 3 white and 3 red faces. I couldn't help myself so I began to play with them... And by play with them, I mean that I began to make a math problem about how many arrangements I could make with them. So here it is:

When lined up side by side and only looking at the top face of each cube. How many different arrangements of the identical cubes can I make such that there is no \(180^{\circ}\) rotational symmetry along the x,y, or z axis or around the origin.


The origin is the center of the 4x1x1 rectangular prison formed by aligning the four blocks.

If the letters have the same orientation and order as another combination but has a different color order, it's considered different and does not have reflectional symmetry.

Hint: if the top of the four blocks are in this specific order \(\boxed{\lfloor}\boxed{\lceil}\boxed{\rfloor} \boxed{\rceil}\) it is rotationally symmetric Around the y axis.


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