The above wall is acceptable. But the configuration below is not acceptable, because the enclosed area consists of two separate "rooms", and because not all 24 blocks are used:
Task: Design your own original enclosure and show it to others by drawing a plan on the blackboard or showing it on grid paper on an overhead. (You won't run out of designs!)
Questions:
What is the biggest area that can be enclosed? (25 squares)
What is the smallest area? (9 squares)
The configuration below is not a legal example of an enclosure with smallest area, because
1. It has two separated rooms, and
2. Not all cubes are connected to 2 other cubes
How many designs have these properties?
(The biggest one: only one. The smallest one: very many.)
Now let's count how many faces form the inside part of the wall, and how many form the outside part of the wall. 20 faces are inside, and 28 are outside. What are the other possibilities?
Answer:
There are no other possibilities; it is always 20 inside and 28 outside.
Proof
Walk on top of the wall all the way around, going clockwise. On each "outside" corner you have to turn right 90 degrees. On each inside corner you have to turn left 90 degrees. In order to return to the original position, you have to make 4 more right turns than left turns.
Now count:
Each cube contributes 2 faces to the total number of inside and outside faces.
Total = 2*24 = 48 faces.
Each outside corner contributes 2 faces to the outside and no faces to the inside.
Each inside corner contributes no faces to the outside, but 2 to the inside.
Other cubes contribute 1 to the outside and 1 to the inside.
Therefore:
Outside + Inside = Total = 48.
Outside - Inside = 2*4 = 8
(4 is the difference between the number of right and left turns, which is also the difference between the number of outside and inside corners.)
The solution to these two equation yields Outside = 28, and Inside = 20.
Remarks:
Do not expect that students can "discover" this proof. But some may understand it.
There are other proofs. But all proofs that we know seem to be equally complex.
Lesson Index