Difference between revisions of "2023 AMC 10A Problems/Problem 18"
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Since each <math>3</math>-point intersection has <math>3</math> adjacent rhombuses, we know the number of <math>3</math>-point intersections must equal the number of <math>3</math>-point intersections per rhombus times the number of rhombuses over <math>3</math>. Since there are <math>12</math> rhombuses and <math>2</math> <math>3</math>-point intersections per rhombus, this works out to be: | Since each <math>3</math>-point intersection has <math>3</math> adjacent rhombuses, we know the number of <math>3</math>-point intersections must equal the number of <math>3</math>-point intersections per rhombus times the number of rhombuses over <math>3</math>. Since there are <math>12</math> rhombuses and <math>2</math> <math>3</math>-point intersections per rhombus, this works out to be: | ||
− | + | \frac{<math>2*12</math>}{<math>3</math>} | |
Hence: <math>\fbox{(D) 8}</math> | Hence: <math>\fbox{(D) 8}</math> |
Revision as of 20:31, 9 November 2023
Contents
[hide]Problem
A rhombic dodecahedron is a solid with congruent rhombus faces. At every vertex,
or
edges meet, depending on the vertex. How many vertices have exactly
edges meet?
Solution 1
Note Euler's formula where . There are
faces and the number of edges is
because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are
vertices on the figure. Let
be the number of vertices with degree 3 and
be the number of vertices with degree 4.
is our first equation. Now note that the sum of the degrees of all the points is twice the number of edges. Now we know
. Solving this system of equations gives
and
so the answer is
.
~aiden22gao ~zgahzlkw (LaTeX)
Solution 2
With 12 rhombi, there are sides. All the sides are shared by 2 faces. Thus we have
shared sides/edges.
Let be the number of edges with 3 vertices and
be the number of edges with 4 vertices.
We get
.
With Euler's formula,
.
, so
. Thus,
.
Solving the 2 equations, we get
and
.
Even without Euler's formula, we observe that a must be even integers, so trying even integer choices and we also get .
Or with a keener number theory eye, we mod 4 on both side, leaving
mod
. Thus, x must be divisible by 4.
~Technodoggo ~zgahzlkw (small edits)
Solution 3
Note that Euler's formula is . We know
from the question. We also know
because every face has
edges and every edge is shared by
faces. We can solve for the vertices based on this information.
Using the formula we can find:
Let
be the number of vertices with
edges and
be the number of vertices with
edges. We know
from the question and
. The second equation is because the total number of points is
because there are 12 rhombuses of
vertices.
Now, we just have to solve a system of equations.
Our answer is simply just
, which is
~musicalpenguin
Solution 4
Each of the twelve rhombuses has two pairs of angles across from each other that must be the same. If both pairs of angles occur at -point intersections, we have a grid of squares. If both occur at
-point intersections, we would have a cube with six square faces. Therefore, two of the points must occur at a
-point intersection and two at a
-point intersection.
Since each -point intersection has
adjacent rhombuses, we know the number of
-point intersections must equal the number of
-point intersections per rhombus times the number of rhombuses over
. Since there are
rhombuses and
-point intersections per rhombus, this works out to be:
\frac{
}{
}
Hence: