To solve this problem, let's go through it step-by-step.

Step 1: Identify the Face with Numbers 8 and 505

The cube is an $8 \times 8 \times 8$ structure, with 512 small cubes (since $8^3 = 512$). Each layer is numbered sequentially with 64 cubes per layer, where:

• First Layer (1–64): Numbered in a zig-zag manner as described.
• Second Layer (65–128): Above the first layer, similarly numbered.

The face with numbers 8 and 505 is on opposite ends of the cube:

• Number 8 is the last number of the first row in the first layer.
• Number 505 is in the last layer (eighth layer) in a corresponding corner position.

Thus, the face containing 8 and 505 is one of the vertical faces of the cube.

Step 2: Determine the Face Diagonal on This Face

The face diagonal is the line connecting opposite corners of a face on the cube. Since 8 and 505 are in opposite corners, the face diagonal would include these two numbers.

Step 3: Sum the Numbers on the Face Diagonal

The numbers on the face diagonal connecting 8 to 505 form a sequence from the first layer to the eighth layer in a straight line along the face of the cube.

The sequence for the face diagonal will include one number from each layer:

• Layer 1: 8
• Layer 2: 72
• Layer 3: 136
• Layer 4: 200
• Layer 5: 264
• Layer 6: 328
• Layer 7: 392
• Layer 8: 456

Now, we calculate the sum of these numbers:

$8 + 72 + 136 + 200 + 264 + 328 + 392 + 456 = 1856$

Conclusion

The sum of the numbers on the face diagonal of the cube face having numbers 8 and 505 is 1856.

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Would you like more details or have any questions?

Here are five related questions you might explore:

1. How would the numbering change if the cube were numbered top to bottom instead of bottom to top?
2. What is the sum of all numbers on the diagonals of a layer?
3. If additional faces are painted in blue, how many more cubes would be blue?
4. How many cubes in total are adjacent to any cube painted blue?
5. How does the arrangement of numbers change if the rows within each layer are numbered identically?

Tip: For large grids, breaking down the structure layer-by-layer can simplify identifying patterns and sequences.