How many distinct ways are there to label the faces of a cube with distinct numbers from 1 to 6, such that there is at least one pair of opposite faces which do not sum to 7?
Rotations (which preserve orientation) are considered the same way.
Reflections (which do not preserve orientation) are considered distinct.
To find the number of distinct ways to label the faces of a cube with distinct numbers from 1 to 6 such that there is at least one pair of opposite faces that do not sum to 7, we can break the problem down into three cases:
Case 1: Both pairs of opposite faces have numbers that do not sum to 7.
Case 2: Only one pair of opposite faces has numbers that do not sum to 7.
Case 3: All pairs of opposite faces have numbers that sum to 7.
Let's analyze each case separately:
Case 1: Both pairs of opposite faces have numbers that do not sum to 7.
In this case, we have two pairs of opposite faces that cannot sum to 7. We need to consider all possible arrangements of numbers that fall into this category. One way to approach this is to fix one face's number, and then choose the numbers for the neighboring faces (taking into account that they cannot sum to 7). There are 6 choices for the fixed face, and once it is chosen, there are 4 choices for the neighboring face since it cannot be the number that, when added to the fixed face, sums to 7. Once the neighboring face is chosen, there are 3 choices for the remaining faces since they need to be distinct from each other and from the fixed and neighboring faces. Thus, there are 6 * 4 * 3 * 2 * 1 = 144 possible arrangements in this case.
Case 2: Only one pair of opposite faces has numbers that do not sum to 7.
In this case, we have one pair of opposite faces that cannot sum to 7, while the other two pairs must sum to 7. We can approach this by fixing the numbers for the pair that cannot sum to 7 and then assigning numbers to the remaining faces. There are 6 choices for the fixed pair of faces, and once they are chosen, there are 2 choices for the numbers in the opposite pair since they must sum to 7. For the remaining two faces, there are 4 choices (excluding the numbers already used). Thus, there are 6 * 2 * 4 * 3 = 144 possible arrangements in this case.
Case 3: All pairs of opposite faces have numbers that sum to 7.
In this case, all opposite pairs of faces must sum to 7. We need to consider all possible arrangements of numbers that fall into this category. One way to approach this is to fix one face's number and the number on its opposite face. There are 6 choices for the fixed face, and once it is chosen, there is only 1 choice for the opposite face since they must sum to 7. For the remaining faces, there are 4 choices (excluding the numbers already used). Thus, there are 6 * 1 * 4 * 3 * 2 * 1 = 144 possible arrangements in this case.
To get the total number of distinct ways to label the faces of a cube, we add up the results from each case: 144 + 144 + 144 = 432.
Therefore, there are 432 distinct ways to label the faces of a cube with distinct numbers from 1 to 6, such that there is at least one pair of opposite faces which do not sum to 7.