Prism of Cubes Puzzle: What will be the least number of exposed cube faces?
What will be the least number of exposed cube faces when 500 cubes of side length 1 cm each are glued together in the form of a prism?
A number of cubes of side length 1 cm each are glued together in the shape of a rectangular prism. After the prism is ready, you can pick it up and see its all faces. If 500 such cubes are used in making the rectangular prism, what will be the minimum number of exposed cube faces?
Solution thoughts
How to get deal with this 3 dimensional prism puzzle mathematically?
- It needs a conversion or of a 3-D image based puzzle to a number based problem that you can deal with easily using simple mathematical concepts.
The prism is uniquely identified by its length, breadth and thickness, all in cm, as side length of a cube is 1 cm. This is the first step of conversion. Let us call the length as L, breadth as B and thickness by T.
Total exposed number of faces of the prism will be its total surface are in sq cm, again because of side length of cube 1 cm.
So we have to find the minimum possible surface area when the prism is made of 500 cubes.
Surface area of the prism
Mathematically surface area is 2(LB + BT + LT).
We have to minimize this.
Let us take an example of an arrangement of a 500-cube prism with 125 cubes along its length, 2 along its breadth and 2 its thickness. Total is 125 x 2 x 2 = 500.
Its area is 2(125 x 2 + 125 x 2 + 2 x 2) = 1008, a very large number of exposed faces of cubes! This is the first experimental configuration to learn what kind of shape has how many exposed faces.
Why is the number of faces so large?
- It is because its long length compared to its thickness and breadth failed to hide the exposed faces that are inside the prism and not visible.
Our mathematical shape logic says,
- The denser we make the prism, more will the hidden faces be and less will its exposed area be.
To make the prism densest is easy. Choose a set of length, breadth and thickness all as near to each other in value.
To do this systematically and confidently, we have to first breakup the number 500 in all its smallest prime factors: 500 = 2 x 2 x 5 x 5 x 5. These are its 5 factors.
- We need to combine these 5 factors is three groups in such a way that the three numbers of factor products will be of nearest value to each other.
- By observation, while mathematically comparing the values, we decide, the densest prism will have length as 10, bread as 10 and thickness as 5.
Area of this prism is: 2(10 x 10 + 10 x 5 + 10 x 5) = 400.
This should be the least number of exposed faces prism configuration.
Other configuration prisms
What about other combinations of L, B and T?
Third configuration: 20, 5, 5:
Its area 2(20 x 5 + 20 x 5 + 5 x 5) = 500, much larger, as expected.
The fourth configuration: 25, 10, 2:
Its area 2(25 x 10 + 10 x 2 + 25 x 2) = 620, larger than the fourth configuration simply because the length, breadth and thickness are spread wider.
There will be another configuration of 50, 5, 2 with area 2(50 x 5 + 5 x 2 + 50 x 2) = 720. The area is increasing as the three values are spread wider from each other.
This confirms our problem solving rule: the least area prism of 400 exposed faces will be the densest with closest length, breadth and thickness.
Forming this general problem solving rule for this type puzzle has its own importance.
Food for thought: What will be the least number of exposed faces prism made up of 28000 cubes? Will you compare areas of each possible configuration? Or apply your new-found problem solving rule?
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