Brilliantia's Buildings
On the island of Brilliantia , the basic building element is a woven wall of fronds supported at each end by a vertical pole. A single pole may support a number of walls radiating from it. A room is any continuous area completely surrounded by walls. A collection of connected walls makes up a building.
How many buildings are on the island if 345 walls, supported by 234 poles, surround 123 rooms?
Details and Assumptions:
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Every wall must be supported on both ends, but this can be done by a single pole.
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Walls may not cross each other.
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A building which is completely surrounded by another building but is not attached to it counts as a separate building.
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A building does not need to contain any rooms. A standalone wall, for example, counts as a building.
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Outdoors does not count as a room.
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A pole may not stand alone.
For example in the above diagram there are 14 walls, 13 poles, 3 rooms, and 2 buildings.
The answer is 12.
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1 solution
Outline of the calculation:
For every building the number of poles, P , the number of rooms, R , and the number of walls, W , satisfy the equation:
P + R − W = 1
So since overall P + R − W = 2 3 4 + 1 2 3 − 3 4 5 = 1 2 , this can only happen if there are 1 2 buildings.
Derivation of the equation:
Any building can start only by first putting up a single wall. There are two kinds of walls. A wall supported by two poles will have P = 2 , R = 0 , W = 1 . A wall supported by just one pole will have P = 1 , R = 1 , W = 1 . So in either case there is just one wall for two items of the other kind, and P + R − W = 1 .
When expanding a building with a new wall-
(1) we can attach the new wall on both ends to a single existing pole. This generates one new room, no new poles, and one new wall. So the value of P + R − W remains unchanged.
(2) we can attach the two ends of the new wall to two existing poles. This generates one new room, no new poles, and again one new wall. P + R − W again remains unchanged.
(3) we can attach just one end of the new wall to an existing pole leaving the other on a new pole. This generates no new rooms, one new pole, and one new wall. So P + R − W again remains unchanged.
We could put the wall up completely separately, but that would be starting a new building.