A wooden block has dimensions 6 cm by 4 cm by 1 cm. The two faces measuring 6 cm x 4 cm are painted green, the two faces measuring 6 cm x 1 cm are painted red, and the two faces measuring 4 cm x 1 cm are painted black. The block is cut into 24 unit cubes of side 1 cm. How many of these small cubes have green paint on exactly one face and no paint on their other five faces?

Introduction / Context:
This is a three dimensional geometry and counting puzzle. You start with a rectangular wooden block whose faces are painted in different colours: two opposite large faces are green, two long thin faces are red, and two short thin faces are black. The block is then cut into 1 cm cubes. The question asks how many of these small cubes show green paint on exactly one face and have no paint on the remaining five faces. Problems like this test spatial imagination and careful reasoning about which smaller cubes lie on edges, corners, or strictly inside larger painted faces.

Given Data / Assumptions:

• The block measures 6 cm in length, 4 cm in breadth, and 1 cm in height.
• The two 6 cm x 4 cm faces are painted green.
• The two 6 cm x 1 cm faces are painted red.
• The two 4 cm x 1 cm faces are painted black.
• The block is cut into 1 cm cubes, so there are 6 * 4 * 1 = 24 small cubes.
• We want cubes that have exactly one green face and all other five faces unpainted.

Concept / Approach:
The key idea is to understand which positions in the block give cubes one green face but keep them away from the edges and corners. Every cube that lies on a green face will have at least one green face. However, cubes at the corners or along the edges where green meets red or black will have extra painted faces in red or black. Only those cubes that are on a green face, but not on any edge of that face, will have exactly one face painted green and no other colour on the remaining faces. Because the height is only 1 cm, the block has a single layer of cubes between the two green faces.

Step-by-Step Solution:
Step 1: Visualise the block as a 6 by 4 grid of 1 cm cubes on the top green face and another 6 by 4 grid on the bottom green face. Step 2: Focus first on the top green face. All 24 cubes belong to either the top face or the bottom face, since the height is 1 cm. Step 3: On the top green face, corner cubes have three painted faces (green plus one red plus one black) and edge cubes have two painted faces (green plus either red or black). Step 4: The only cubes with exactly one green face on the top layer are the interior cubes of the 6 by 4 rectangle, which do not touch any red or black sides. Step 5: Count those interior cubes on the top face by excluding the outer border. Along the length, exclude the two end columns, leaving 6 - 2 = 4 interior columns. Along the breadth, exclude the two border rows, leaving 4 - 2 = 2 interior rows. Step 6: The interior count on the top face is 4 * 2 = 8 cubes, each having exactly one green face and no other paint. Step 7: Repeat the same reasoning for the bottom green face. It has the same pattern, giving another set of 8 interior cubes with exactly one green face. Step 8: Add the two sets: 8 from the top face plus 8 from the bottom face gives 16 cubes.
Verification / Alternative check:
As a check, consider the total structure. There are 24 cubes. Corners and edges on the top and bottom faces account for those cubes that touch red or black faces. Specifically, on each green face there are 4 corner cubes and 8 edge cubes that are not corners, making 12 boundary cubes per face. Because the block is only one cube high, each physical corner or edge cube on the top has the same cube as boundary on the bottom. Counting carefully, all boundary cubes have more than one painted face, and the remaining 16 interior positions across the two green faces are the only ones that have exactly one green face and no other paint.

Why Other Options Are Wrong:
An answer of 8 would count only one green face, forgetting that there are two such faces. An answer of 24 would assume that every cube with green paint has only one green face, which ignores edges and corners where other colours appear. An answer of 4 is too small and would correspond only to a subset of interior cubes, not the full count from both faces. Only 16 matches the full interior positioning on both green faces.

Common Pitfalls:
A common error is to think only about the top green face and then stop, forgetting that the bottom face also contributes an equal number of cubes. Another pitfall is to miscount cubes on edges and corners, accidentally including cubes that have additional red or black paint. Drawing a simple 6 by 4 grid and marking border and interior positions is a reliable way to avoid miscounting and to see clearly which cubes qualify.

Final Answer:
There are 16 small cubes that have green paint on exactly one face and no paint on their other five faces.