Folded Cube Face Positions
Imagine a flat net made of six equal squares. Four squares form a vertical strip, one above the other. Label these four squares, from top to bottom, P, Q, R, and S. A fifth square, labelled T, is attached to the left edge of Q. A sixth square, labelled U, is attached to the right edge of Q. The net is folded along the edges of the squares to make a cube, with the labels on the outside. No square overlaps another, and every square becomes one face of the cube. You may find it useful to track which faces end up opposite each other and how the cube can be rotated. The cube is then placed on a table. In the first position, face P is on top and face T is facing you. In a second position, face U is on the bottom and face S is facing you.
1. Which face is opposite U? Explain by describing the folding.
Read the worked answer
T is opposite U. Fold Q as the front face. T, attached to the left of Q, folds to become the left face. U, attached to the right of Q, folds to become the right face. The left and right faces are opposite each other. Equivalently, the vertical strip P–Q–R–S gives P opposite R and Q opposite S; the only remaining pair is T and U.
2. In the first position (P top, T facing you), which face is on the bottom and which face is on your right?
Read the worked answer
Bottom: R. Right: Q. P is opposite R, so if P is on top, R must be on the bottom. In the standard folded cube, with Q at the front, T is on the left and U is on the right. To bring T to the front, rotate the cube 90° about the vertical axis so that the left face comes to the front. The old front face Q then moves to the right. P stays on top and R stays on the bottom. So the right-hand face is Q.
3. In the second position (U bottom, S facing you), which face is on top and which face is on your right?
Read the worked answer
Top: T. Right: R. U is opposite T, so if U is on the bottom, T must be on top. S is opposite Q, so if S is facing you, Q is at the back. The remaining side faces are P and R. To decide which is on the right, start from the standard position (Q front, S back, U right, T left, P top, R bottom) and rotate so that U goes to the bottom. A 90° roll about the front–back axis sends U to the bottom, P to the right, R to the left, and T to the top. Now S is still at the back. Rotate the cube 180° about the vertical axis to bring S to the front. This swaps left and right: P goes to the left and R goes to the right. Therefore, with U bottom and S front, the right-hand face is R.