A solid wooden block has dimensions 5 cm by 10 cm by 20 cm. Identical blocks are to be arranged to form a solid wooden cube of the smallest possible size, with no gaps. How many such blocks are required to construct this cube?

Introduction / Context:
This problem involves assembling a three dimensional cube from identical rectangular blocks. Each block has dimensions 5 cm by 10 cm by 20 cm. Our goal is to construct the smallest possible cube whose dimensions are compatible with these blocks. The key idea is that the cube side must be a common multiple of the three given dimensions, and the arrangement of blocks must fill the cube exactly with whole blocks, no cutting and no gaps. This is a typical puzzle style question that tests spatial reasoning and understanding of least common multiples in three dimensions.

Given Data / Assumptions:

• Block dimensions: 5 cm, 10 cm and 20 cm.
• We can orient blocks in any direction, but they must fit exactly.
• The final solid is a cube, so all three sides of the final shape must be equal.
• We want the cube of minimum possible size.

Concept / Approach:
Let the side of the cube be L centimetres. For the cube to be filled exactly, along each of its three dimensions, we must be able to fit an integer number of block edges, possibly matching 5, 10 or 20 along each direction depending on orientation. The smallest such cube will have side length equal to the least common multiple of 5, 10 and 20. Once L is known, we compute the volume of the cube, divide by the volume of one block, and obtain the number of blocks required.

Step-by-Step Solution:
Step 1: Find the least common multiple (LCM) of the block edges. Edges are 5, 10 and 20 centimetres. LCM of 5, 10 and 20 is 20 centimetres. Thus the smallest possible cube side length L = 20 centimetres. Step 2: Arrange blocks to fit the cube. We can orient each block so that its longest side (20 cm) lies along one cube edge. Then along that edge, only one block fits: 20 / 20 = 1. The 10 cm side can align with another cube edge, giving 20 / 10 = 2 blocks along that direction. The 5 cm side can align with the remaining cube edge, giving 20 / 5 = 4 blocks along that direction.
Step 3: Compute number of blocks. Number of blocks along three dimensions: 1, 2 and 4. Total number of blocks = 1 * 2 * 4 = 8. Thus 8 blocks are required to form the smallest solid cube.

Verification / Alternative check:
Volume of each block = 5 * 10 * 20 = 1000 cubic centimetres. Volume of final cube with side 20 cm = 20^3 = 8000 cubic centimetres. Number of blocks = 8000 / 1000 = 8, which confirms our earlier count. The cube is completely filled without gaps, and the side length is indeed the smallest multiple that accommodates all block dimensions.

Why Other Options Are Wrong:
Option A (6), Option C (12), Option D (16), and Option E (24) correspond to numbers of blocks that would either not make a perfect cube or require a cube with a side larger than 20 cm. None of these counts match the exact volume ratio or a consistent integer arrangement in three dimensions for the smallest cube.

Common Pitfalls:
Some learners attempt to use the greatest common divisor instead of the least common multiple or forget that the final shape must be a cube. Others might try to force a smaller cube, which is impossible given the dimensions. The safest method is to use the least common multiple to determine the cube side and then verify by volume division.

Final Answer:
The minimum number of such blocks required to form a solid cube is 8.