Three different positions X, Y and Z of a die are shown in the figures given…

When the same die is shown in several different positions, any number that appears as a visible face in two different views is adjacent to every number that appeared alongside it in each of those views. Since every face of a cube touches exactly four other faces, once four distinct numbers have been identified as adjacent to a given number, the one remaining number (out of the six) must be the number on its opposite face.

1. In position X, the visible faces are 5 (top), 4 and 1. In position Y, the visible faces are 4 (top), 6 and 3. In position Z, the visible faces are 2 (top), 6 and 5.

2. The number 4 is visible in position X (alongside 5 and 1) and in position Y (alongside 6 and 3). So 4 is adjacent to 5, 1, 6 and 3 -- four distinct numbers. The only number from 1-6 not among these four (and not 4 itself) is 2, so 2 must lie opposite 4.

3. The number 6 is visible in position Y (alongside 4 and 3) and in position Z (alongside 2 and 5). So 6 is adjacent to 4, 3, 2 and 5 -- four distinct numbers. The only remaining number is 1, so 1 must lie opposite 6.

4. With 4 paired opposite 2, and 6 paired opposite 1, only two numbers remain unpaired: 3 and 5. They must therefore lie opposite each other.

5. In position X, the top face shows 5. Since the face opposite 5 has now been fixed, that is the number occupying the bottom face of position X.

As an independent check, opposite numbers should never appear together in the same view: 4 and 2 never occur in the same position (X and Y show 4 but not 2, Z shows 2 but not 4); 6 and 1 never occur together (Y and Z show 6 but not 1, X shows 1 but not 6); and 3 and 5 never occur together across X, Y or Z either. This confirms the pairing is consistent with all three given positions.

Therefore, the bottom face of position X shows the number that lies opposite 5 in this pairing.