# What is an opposite face to the letter $T$?

Starting with 27 small cubes connected to each other in a 3x3x3 cube shape, I removed some of the cubes so that all the remaining cubes stayed solidly connected by cube faces.

I then used the resulting object as a stamp: I touched one face in ink and pressed it on paper. After doing this for 3 faces of the object, I had managed to spell out the word TOY:

X X X    X X X    X   X
X      X   X      X
X      X X X      X


Question 1. (the original question): Which letter can be stamped using the face that's opposite to the O face?

Question 2. The face opposite the T face can also be used to stamp a clearly recognisable letter. Which one?

(nb. The question didn't originally specify that the small cubes had to fit inside a 3x3x3 volume and be connected by cube faces.)

• I had trouble making sense of this question. It is not well-defined. You should say that you are creating a stamp from the cubes, and that you are doing so by removing some of the cubes in the 3x3x3 cube. There are too many assumptions you need to do before you can solve it. Aug 16, 2020 at 11:07
• Since this is quite a nice puzzle, I did my best in rewriting the description to be as readable and well-defined as I could. Please feel free to fix or revert the edit if there's anything you don't like about it.
– Bass
Aug 16, 2020 at 17:27
• True. It is a nice puzzle. The problem was only in the way it was asked. Aug 18, 2020 at 14:41

Answer to the original question "What pattern can you print with the face opposite the O?":

You can make

the letter Y.

That would be because in a 3x3x3 cube

neighbouring faces always share exactly three cubies along an edge:

(image is from https://commons.wikimedia.org/wiki/File:Rubiks_cube.svg)

The O face (I see what you did there!) has three cubies along every edge, so any neighbouring side must have that, too.

Since the face with the Y has no such edges,

it cannot be adjacent to the O face.

Answer to the question that OP edited in later:

Making the bold assumption that the side opposite the T also prints a clearly recognisable letter, it's going to be

The letter U.

That would be because the side opposite the T must be one of these two:

 X X X      X X X
X X X      X   X
X   X      X   X


There are no other possibilities, because there's only one way the T side can connect to the O and Y sides, and the corner pieces on the Y side need to be connected face-wise to the rest of the shape. (They aren't connected on the Y side.)

To see this clearly, here's a 3D model of the three known sides:

• I have edit the question after your answer. I have changed the O face to the T face in the question.
– Nick
Aug 16, 2020 at 8:36
• Excellent work, but I think you should add two cubes in your model. Because, the answer is the U letter.
– Nick
Aug 16, 2020 at 10:09
– Bass
Aug 16, 2020 at 10:19

As far as 3x3 gridded shapes go:

anything you like (except an empty grid)

For example an "X"

For other shapes you use essentially the same pattern slightly retracting the pixels that are not wanted. The top left corner has to be special cased if it is not wanted it has to be left out entirely. If we are required to use up all cubes it is easy to hide it inside the shape.

• Why is the blue centre cube in middle?
– Nick
Aug 16, 2020 at 7:37
• @Nick because I wasn't paying attention. FIxed now. Aug 16, 2020 at 7:39
• The O face has three cubies along every edge, so the T face should be rotated.
– Nick
Aug 16, 2020 at 8:29
• @Nick Now that you've edited the question, of course, everything changes. I'll stick with the original problem and leave the evolving question for others to answer. Aug 16, 2020 at 8:38
• Sorry, for that, but I see now that is trivial one in the original problem.
– Nick
Aug 16, 2020 at 8:45