To solve the problem:

1. Dimensions of the box: 685 mm (length), 411 mm (width), 411 mm (height).
2. The goal is to find the largest cube size that can fit evenly into the box without leaving any space.

Step 1: Determine the largest cube size

To find the largest cube size, we compute the greatest common divisor (GCD) of the three dimensions:

• Dimensions: 685, 411, 411.

Finding the GCD:

1. GCD(685, 411) = 137
2. GCD(137, 411) = 137

Thus, the largest cube size is 137 mm.

Step 2: Number of cubes in the box

The number of cubes along each dimension:

1. Length: $\frac{685}{137} = 5$
2. Width: $\frac{411}{137} = 3$
3. Height: $\frac{411}{137} = 3$

Total number of cubes: $5 \times 3 \times 3 = 45$

Final Answer:

• The size of each cube is 137 mm.
• The total number of cubes that fit in one box is 45.

Let me know if you need further clarification or details!

Related Questions:

1. How is the GCD used in determining cube sizes for such problems?
2. What happens if the dimensions are not divisible by the same number?
3. Could a rectangular prism be used instead of a cube to optimize filling?
4. How would the result change if one box dimension were slightly altered?
5. How can this concept apply to 3D packing in logistics?

Tip:

Always check for the greatest common divisor (GCD) when fitting identical units into a larger space!