A different letter is painted on each face of a cube. This cube is shown below in 3 different positions:
![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-086.jpg?height=154&width=452&top_left_y=1278&top_left_x=824)

What letter belongs on the shaded face of this cube in the following diagram?
(A) $T$
(B) $P$
(C) $X$
(D) $E$
(E) $V$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-086.jpg?height=151&width=138&top_left_y=1456&top_left_x=1579)
We will determine which letter belongs on the shaded face by "unfolding" the cube.

Using the first of the three positions, we obtain
![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-368.jpg?height=112&width=112&top_left_y=1865&top_left_x=1082)

Using the third of the three positions, we can add the "A" above the "E" as

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-368.jpg?height=171&width=127&top_left_y=2034&top_left_x=1476)

Using the second of the three positions, the F can be placed above the A and the $\mathrm{X}$ to the left of the $F$ to get

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-368.jpg?height=225&width=179&top_left_y=2294&top_left_x=518)

If we refold this cube, with $\mathrm{A}$ at the front, $\mathrm{F}$ on top, and $\mathrm{E}$ on the bottom, the right-hand face will be the $\mathrm{V}$ (upside down), and so the shaded face is the V.