Seven identical playing dice are unfolded in an identical manner. Six of the unfolded dice are laid out on a flat surface and are arranged to touch each other without overlapping. The figure shows a portion of the arrangement. Which face(s) from the seventh unfolded dice CAN NOT replace X if overlap must be avoided?
To solve this problem, we need to determine which face(s) from the seventh unfolded die cannot be used to replace the face marked X without causing overlap in the given arrangement. A standard die has a fixed arrangement of numbers on its faces: opposite faces always sum up to 7. Let's break down the steps:
Identify the faces of a standard die:
1 is opposite 6
2 is opposite 5
3 is opposite 4
Based on these pairings, any face adjacent to X on an arranged die will be one of the adjacent numbers and not the opposite. For instance, if X is shown as face 1, its adjacent faces must be 2, 3, 4, or 5. Face 6, being opposite to 1, cannot be adjacent.
By observing the arrangement, if we cannot add X as face 1, it means none of the faces directly opposite those in contact with existing dice can be placed. Repeat this logic for each face that has been used.
Based on typical layouts and the problem's requirements, faces that correctly align would be all but the directly opposing face aligning with 6, given X is face 1. This means face 6 can't be used.
Therefore, the faces that CANNOT replace X are those directly opposite existing faces within the arrangement, based on identical face pairings of a die. Specifically, the faces that cannot replace X based on the conditions provided are those shown above.
