2009 AMC 12B Problems/Problem 20
A convex polyhedron has vertices , and edges. The polyhedron is cut by planes in such a way that plane cuts only those edges that meet at vertex . In addition, no two planes intersect inside or on . The cuts produce pyramids and a new polyhedron . How many edges does have?
Each edge of is cut by two planes, so has vertices. Three edges of meet at each vertex, so has edges.
At each vertex, as many new edges are created by this process as there are original edges meeting at that vertex. Thus the total number of new edges is the total number of endpoints of the original edges, which is . A middle portion of each original edge is also present in , so has edges.
Euler's Polyhedron Formula applied to gives , where F is the number of faces of . Each edge of is cut by two planes, so has vertices. Each cut by a plane creates an additional face on , so Euler's Polyhedron Formula applied to gives , where is the number of edges of . Subtracting the first equation from the second gives , whence . The answer is .
Each edge connects two points. The plane cuts that edge so it splits into at each end (like two legs) for a total of new edges.
But because each new edge is shared by an adjacent original edge cut similarly, the additional edges are overcounted .
Since there are edges to start with, new edges result. So there are edges in the figure.
The question specifies the slices create as many pyramids as there are vertices, implying each vertex owns 4 edge ends. There are twice as many edge-ends as there are edges, and .
, so there are vertices.
The base of a pyramid has 4 edges, so each sliced vertex would add four edges to .
I doubt anyone has the mental capacity to imagine in its full complexity in their mind. So, we begin by examining a simpler structure: a cube.
Let denote the number of edges of a polyhedron . Let denote the number of vertices of . Let denote the number of edges that meet at each vertex.
Suppose is a cube. Then
, , and
(A cube has edges, vertices, and edges meet per vertex)
Why? Well, imagine an ant walking along an edge of a cube. When it reaches the end of the edge, it will be standing on a vertex (by definition). But, what if it had walked the other way? Well, it would still be on the same edge, but it would reach the other endpoint of the edge, another vertex. Therefore, we can say that a polyhedron with edges has total endpoints. In the case of , a cube, this means a cube has total endpoints. But a cube only has vertices, not ... This doesn't make any sense... Wait a minute, we're dealing with a polyhedron, which means multiple edges will be connected to the same vertex. In the case of a cube, edges meet at every vertex. So we have , or vertices. The math works out. Let's apply this to , our mystery polyhedron.
We are told that , and isn't given explicitly. So we'll let . Now our equation reads
What does this mean, exactly? Well has edges, so it must has endpoints (endpoints, not vertices). So if we can figure out either , the number of vertices, or , the number of edges per vertex, we can figure out the other variable. Hmmmm... The problem doesn't give us any clues about either. How frustrating. Let's leave this alone and consider the other information given to us.
We are also told that "the polyhedron is cut by planes " (at this point I imagine is a giant block of cheese, potato, etc.), and also that "plane cuts only those edges that meet at ". What does this mean?
Well, let's go back to our cube. Notice that there are exactly as many planes as vertices. Think of the planes as knives (or knife strokes) cutting into a big cube of cheese. We're cutting the cheese times, since there are vertices. Also, whenever we cut into the cheese, we have to direct the cuts near a vertex, cutting into only the edges connected to that vertex. How do we do that? Well, since we can't cut into any other edges, we're gonna have to settle for cutting corners (see what I did there? ...Ok, I admit it was cheesy... see what I did there, too?😂). We have to shave off just the corners of the cheese block, because this way it won't interfere with any of the other edges.
The problem also tells us that none of the planes overlap. In our cheese analogy, this means that every time we cut, we have to make sure the cuts are always into new cheese, and not into previously cut areas (all of the cuts are distinct, and don't overlap). Sure enough, this is in agreement with the information that exactly pyramids are cut off. Let's go back to .
has vertices, so we're gonna make cuts. The question we want to answer is: "After all of the cuts are made, how many total edges will (the figure that remains), have?". Well, we're gonna have to figure out either or ... or do we? We're told that pyramids were made, but it doesn't specify exactly how many sides the pyramid has. So... that implies that it doesn't matter, or else the problem wouldn't be solvable. Let's suppose ( edges meet at every vertex). Then we would have vertices, and we would make cuts. But how many more edges does it yield? (We are told that the planes don't overlap, so the original edges are all present.) Well, if we cut off the corner of a cube, it will create more edges... which is coincident with the fact that edges meet at every vertex. So for every vertex, we gain exactly new edges. Rewriting the equation to find the number of vertices, we have
But we just said that for every vertex/cut, we gain more edges. So the total number of new edges is . It doesn't matter how many vertices has, or how many edges meet at every vertex. Given the conditions in the problem, we will always gain more edges, regardless of the values of , or . We have new edges old edges to make edges .
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