Question

Shown below is the front and top views of two cubes. The red and blue cubes are made up of metal and wax respectively. Red cube is heated and is hot enough to melt wax in the blue cube. As shown in the front and top views, the red cube is moved along the path PQRS creating sharp edges and flat surfaces on the blue cube. What is the total number of surfaces the blue cube will have after the red cube reaches the point S?

Hint:

Track the path as a sequence of straight cuts across the faces of the cube. Every straight stretch normally exposes new side walls, and remember to include any extra surface exposed exactly where the path changes direction. Count the faces the path never touches separately from the ones it does.

Answer 1

Correct Answer: 16

Step 1: Analyze the given information.
The red cube is moving along the path PQRS, creating sharp edges and flat surfaces on the blue cube as it moves. Initially, the blue cube has 6 surfaces (as it is a cube). Each time the red cube moves along a different part of the blue cube, it modifies the surfaces, either by flattening or exposing new surfaces.
Step 2: Analyze the effect of the red cube movement.
As the red cube moves, it will melt the wax on the blue cube and expose more surfaces of the blue cube. The movement will expose additional surfaces along the path PQRS. This leads to the creation of 16 total surfaces on the blue cube.
Step 3: Conclusion.
Thus, the total number of surfaces the blue cube will have after the red cube reaches point S is \( \boxed{16} \).

Answer 2

Concept:

• A hot object melting its way across a solid does not create surfaces at random - every straight stretch of its path cuts a groove with its own side walls, and every place it changes direction adds one more small surface at that corner.
• Faces the path never reaches stay exactly as they were on the original cube.

Step 1: Count the faces the path never touches.
The red cube travels only over the top and front region of the blue cube along P to Q to R to S, so the left face, right face, back face and bottom face are never touched. That is 4 untouched original faces.

Step 2: Count what is left of the two faces the path does cross.
The path splits the original top face into 2 separate flat pieces, and splits the original front face into 2 separate flat pieces. That gives $2+2=4$ remaining flat pieces from the two faces it crosses.

Step 3: Count the new surfaces created by the groove itself.
The path has 3 straight stretches, P to Q, Q to R and R to S. Each straight stretch melts a groove with 2 new side walls, giving $3\times2=6$ new wall surfaces. The path also changes direction twice, at Q and at R, and each turn exposes 1 extra small corner surface, giving $2\times1=2$ more surfaces. In total the groove itself adds $6+2=8$ new surfaces.

Step 4: Add every surface together.
Untouched faces plus remaining flat pieces plus new groove surfaces $=4+4+8=16$.

Final Answer: 16