10 Observations
1)Each side of the cube has a color, a name, and two numbers.
2)One of the numbers on each side is located in the top right corner of the side, and the other number is located in the bottom left corner of the side.
3)The colors os each side are either red, white, or blue.
4) The names on each side are either a traditionally male or traditionally female name, and each side has a different name.
5)The two blue sides are located on the opposite portion of the cube from each other, the two white sides are located on the opposite portion of the cube from each other, and the red side is located opposite the unknown bottom side.
6)The pairs of sides (as defined by color) contain a male name and a female name respectively, and these names share at least two of the first letters of the name in common.
7)Each (known) pair shares the same number at the bottom left corner with each other.
8)Each visible side has a unique set of two numbers.
9) The number in the top right corner of each visible side is the number of letters in that side's name.
10) The number in the bottom left corner of each side is the number of letters at the beginning of the name that is shared with that side's color pair.
Hypothosis
The bottom of the cube is red, has the number four located in the bottom left corner, the number eight in the top right corner, and is labelled with the name "Francine".
Explanation
We conclude that the bottom of the cube is red because each of the visible sides shares the same color as the opposite side of the cube, and red is the color of the side opposite to the bottom. In each known pair, it is observed that they share the same number in the bottom left corner as their same color opposite side, and as four is the number in the bottom left corner of the opposing (top) side, we conclude that four is the number in the bottom left corner of the bottom side. The top side has a male name label of "Frank", and as we observe the other color pairs contain a male and a female name pair, we conclude that the bottom of the cube has a female name label. We observe in the other pairs that the shared, bottom left number indicates the number of letters at the beginning of both names that are in common, we conclude that the bottom female name must have "Fran" to begin, in order to coincide with the number four located in the bottom left of the top cube and the first four letters of the name "Frank" located on the top side. We deduce that that name is "Francine" as it fulfills the previously laid out requirements, and therefore the number eight is located in the top right corner of the bottom side, as there are eight letters in the name label and each side's top right corner number coincides with the number of letters in the name label.
