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Difference between revisions of "2023 AMC 10A Problems/Problem 18"

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==Solution 1==
 
==Solution 1==
  
−
Note Euler's formula where <math>V+F-E=2</math>. There are <math>12</math> faces and the number of edges is <math>24</math> because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are <math>14</math> vertices on the figure. Let <math>A</math> be the number of vertices with degree 3 and <math>B</math> be the number of vertices with degree 4. <math>A+B=14</math> 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 <math>3A+4B=48</math>. Solving this system of equations gives <math>B = 6</math> and <math>A = 8</math> so the answer is <math>\fbox{D}</math>.
+
Note Euler's formula where <math>\text{Vertices}+\text{Faces}-\text{Edges}=2</math>.  
−
~aiden22gao ~zgahzlkw (LaTeX)
+
There are <math>12</math> faces.
 +
There are <math>24</math> edges, because there are 12 faces each with four edges and each edge is shared by two faces.  
 +
Now we know that there are <math>2-12+24=14</math> vertices.
 +
Now note that the sum of the degrees of all the points is <math>24</math>(the number of edges). Let <math>x=</math> the number of vertices with <math>3</math> edges. Now we know <math>\frac{3x+4(14-x)}{2}=24</math>. Solving this equation gives <math>x = \boxed{\textbf{(D) }8}</math>.
 +
~aiden22gao ~zgahzlkw (LaTeX) ~ESAOPS (Simplified) ~sonic12345 (Fixed typo)
  
−
==Solution 2==
+
==Solution 2 (Cheese)==
  
−
With 12 rhombi, there are <math>48</math> sides. All the sides are shared by 2 faces. Thus we have <math>24</math> shared sides/edges.
+
Let <math>x</math> be the number of vertices with 3 edges, and <math>y</math> be the number of vertices with 4 edges. Since there are <math>\frac{4*12}{2}=24</math> edges on the polyhedron, we can see that <math>\frac{3x+4y}{2}=24</math>. Then, <math>3x+4y=48</math>. Notice that by testing the answer choices, (D) is the only one that yields an integer solution for <math>y</math>. Thus, the answer is <math>\boxed{\textbf{(D) }8}</math>.
  
−
Let <math>A</math> be the number of edges with 3 vertices and <math>B</math> be the number of edges with 4 vertices.
+
~Mathkiddie
−
We get <math>3A + 4B = 48</math>.
 
−
With Euler's formula, <math>V-3+F=2</math>.  <math>V-24+12=2</math>, so <math>V = 14</math>. Thus, <math>a+b= 14</math>.
 
−
Solving the 2 equations, we get <math>a = 8</math> and <math>b = 6</math>.
 
  
−
Even without Euler's formula, we observe that a must be even integers, so trying even integer choices and we also get <math>a = 8</math>.
+
==Solution 3==
−
Or with a keener number theory eye, we mod 4 on both side, leaving <math>3x</math> mod <math>4 + 0 = 0</math>. Thus, x must be divisible by 4.
+
 
 +
With <math>12</math> rhombi, there are <math>4\cdot12=48</math> total boundaries. Each edge is used as a boundary twice, once for each face on either side. Thus we have <math>\dfrac{48}2=24</math> total edges.
 +
 
 +
Let <math>A</math> be the number of vertices with <math>3</math> edges (this is what the problem asks for) and <math>B</math> be the number of vertices with <math>4</math> edges. We have <math>3A + 4B = 48</math>.
 +
 
 +
Euler's formula states that, for all convex polyhedra, <math>V-E+F=2</math>. In our case, <math>V-24+12=2\implies V=14.</math> We know that <math>A+B</math> is the total number of vertices as we are given that all vertices are connected to either <math>3</math> or <math>4</math> edges. Therefore, <math>A+B=14.</math>
 +
 
 +
We now have a system of two equations. There are many ways to solve for <math>A</math>; choosing one yields <math>A=\boxed{\textbf{(D) }8}</math>.  
 +
 
 +
Even without Euler's formula, we can do a bit of answer guessing. From <math>3A+4B=48</math>, we take mod <math>4</math> on both sides.
 +
 
 +
<cmath>3A+4B\equiv48~(\mod{4})</cmath>
 +
<cmath>3A\equiv0~(\mod{4})</cmath>
 +
 
 +
We know that <math>3A</math> must be divisible by <math>4</math>. We know that the factor of <math>3</math> will not affect the divisibility by <math>4</math> of <math>3A</math>, so we remove the <math>3</math>. We know that <math>A</math> is divisible by <math>4</math>. Checking answer choices, the only one divisible by <math>4</math> is indeed <math>A=\boxed{\textbf{(D) }8}</math>.  
  
−
~Technodoggo ~zgahzlkw (small edits)
+
~Technodoggo ~zgahzlkw (small edits) ~ESAOPS (LaTeX)
  
−
==Solution 3==
+
==Solution 4==
 
Note that Euler's formula is <math>V+F-E=2</math>. We know <math>F=12</math> from the question. We also know <math>E = \frac{12 \cdot 4}{2} = 24</math> because every face has <math>4</math> edges and every edge is shared by <math>2</math> faces.  We can solve for the vertices based on this information.  
 
Note that Euler's formula is <math>V+F-E=2</math>. We know <math>F=12</math> from the question. We also know <math>E = \frac{12 \cdot 4}{2} = 24</math> because every face has <math>4</math> edges and every edge is shared by <math>2</math> faces.  We can solve for the vertices based on this information.  
  
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<cmath>f = 6</cmath>
 
<cmath>f = 6</cmath>
 
<cmath>t = 8</cmath>
 
<cmath>t = 8</cmath>
−
Our answer is simply just <math>t</math>, which is <math>\fbox{(D) 8}</math>
+
Our answer is simply just <math>t</math>, which is <math>\boxed{\textbf{(D) }8}</math>
 
~musicalpenguin
 
~musicalpenguin
  
−
==Solution 4==
+
==Solution 5==
−
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 <math>4</math>-point intersections, we have a grid of squares. If both occur at <math>3</math>-point intersections, we would have a cube with six square faces. Therefore, two of the points must occur at a <math>3</math>-point intersection and two at a <math>4</math>-point intersection.
+
Each of the twelve rhombi has two pairs of angles across from each other that must be congruent. If both pairs of angles occur at <math>4</math>-point intersections, we have a grid of squares. If both occur at <math>3</math>-point intersections, we would have a cube with six square faces. Therefore, two of the points must occur at a <math>3</math>-point intersection and two at a <math>4</math>-point intersection.
  
 
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 two <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 two <math>3</math>-point intersections per rhombus, this works out to be:
  
−
\frac{2*12}{3}
+
<math>\frac{2\cdot12}{3}</math>
  
−
Hence: <math>\fbox{(D) 8}</math>
+
Hence: <math>\boxed{\textbf{(D) }8}</math>
 
~hollph27
 
~hollph27
 +
~Minor edits by FutureSphinx
 +
 +
==Solution 6 (Based on previous knowledge)==
 +
Note that a rhombic dodecahedron is formed when a cube is turned inside out (as seen [https://en.wikipedia.org/wiki/Rhombic_dodecahedron#/media/File:R1-cube.gif here]), thus there are 6 4-vertices (corresponding to each face of the cube) and 8 3-vertices (corresponding to each corner of the cube). Thus the answer is <math>\boxed{\textbf{(D) }8}</math>
 +
 +
==Solution 7 (Using Answer Choices)==
 +
Let <math>m</math> be the number of <math>4</math>-edge vertices, and <math>n</math> be the number of <math>3</math>-edge vertices. The total number of vertices is <math>m+n</math>. Now, we know that there are <math>4 \cdot 12 = 48</math> vertices, but we have overcounted. We have overcounted <math>m</math> vertices <math>3</math> times and overcounted <math>n</math> vertices <math>2</math> times. Therefore, we subtract <math>3m</math> and <math>2n</math> from <math>48</math> and set it equal to our original number of vertices.
 +
<cmath>48 - 3m - 2n = m+n</cmath>
 +
<cmath>4m + 3n = 48</cmath>
 +
From here, we reduce both sides modulo <math>4</math>. The <math>4m</math> disappears, and the left hand side becomes <math>3n</math>. The right hand side is <math>0</math>, meaning that <math>3n</math> must be divisible by <math>4</math>. Looking at the answer choices, this is only possible for <math>n = \boxed{8}</math>.
 +
 +
-DEVSAXENA
 +
 +
(Isn't this the same as the last half of Solution 2?)
 +
 +
==Solution 8 (Dual)==
 +
Note that a rhombic dodecahedron is the dual of a cuboctahedron. A cuboctahedron has <math>8</math> triangular faces, which correspond to <math>\boxed{\textbf{(D) }8}</math> vertices on a rhombic dodecahedron that have <math>3</math> edges.
 +
 +
==Video Solution by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/GP-DYudh5qU?si=fFif-OiVZnkdTTv0&t=5105
 +
 +
~Math-X
 +
 +
==Video Solution ==
 +
https://youtu.be/5OuzPFvJPEY
 +
 +
== Video Solution==
 +
 +
https://www.youtube.com/watch?v=Z-OCnHUwnj0
 +
 +
==Video Solution by OmegaLearn==
 +
https://youtu.be/0AG5XmWY-D8
 +
 +
==Video Solution by TheBeautyofMath==
 +
https://www.youtube.com/watch?v=zvKijDeiYUs
 +
 +
==Video Solution==
 +
 +
https://youtu.be/0ssjr8KjOzk
 +
 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 +
 +
==See Also Cheese==
 +
{{AMC10 box|year=2023|ab=A|num-b=17|num-a=19}}
 +
{{MAA Notice}}

Revision as of 20:11, 7 September 2024

Problem

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?

$\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Solution 1

Note Euler's formula where $\text{Vertices}+\text{Faces}-\text{Edges}=2$. There are $12$ faces. There are $24$ edges, because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are $2-12+24=14$ vertices. Now note that the sum of the degrees of all the points is $24$(the number of edges). Let $x=$ the number of vertices with $3$ edges. Now we know $\frac{3x+4(14-x)}{2}=24$. Solving this equation gives $x = \boxed{\textbf{(D) }8}$. ~aiden22gao ~zgahzlkw (LaTeX) ~ESAOPS (Simplified) ~sonic12345 (Fixed typo)

Solution 2 (Cheese)

Let $x$ be the number of vertices with 3 edges, and $y$ be the number of vertices with 4 edges. Since there are $\frac{4*12}{2}=24$ edges on the polyhedron, we can see that $\frac{3x+4y}{2}=24$. Then, $3x+4y=48$. Notice that by testing the answer choices, (D) is the only one that yields an integer solution for $y$. Thus, the answer is $\boxed{\textbf{(D) }8}$.

~Mathkiddie

Solution 3

With $12$ rhombi, there are $4\cdot12=48$ total boundaries. Each edge is used as a boundary twice, once for each face on either side. Thus we have $\dfrac{48}2=24$ total edges.

Let $A$ be the number of vertices with $3$ edges (this is what the problem asks for) and $B$ be the number of vertices with $4$ edges. We have $3A + 4B = 48$.

Euler's formula states that, for all convex polyhedra, $V-E+F=2$. In our case, $V-24+12=2\implies V=14.$ We know that $A+B$ is the total number of vertices as we are given that all vertices are connected to either $3$ or $4$ edges. Therefore, $A+B=14.$

We now have a system of two equations. There are many ways to solve for $A$; choosing one yields $A=\boxed{\textbf{(D) }8}$.

Even without Euler's formula, we can do a bit of answer guessing. From $3A+4B=48$, we take mod $4$ on both sides.

\[3A+4B\equiv48~(\mod{4})\] \[3A\equiv0~(\mod{4})\]

We know that $3A$ must be divisible by $4$. We know that the factor of $3$ will not affect the divisibility by $4$ of $3A$, so we remove the $3$. We know that $A$ is divisible by $4$. Checking answer choices, the only one divisible by $4$ is indeed $A=\boxed{\textbf{(D) }8}$.

~Technodoggo ~zgahzlkw (small edits) ~ESAOPS (LaTeX)

Solution 4

Note that Euler's formula is $V+F-E=2$. We know $F=12$ from the question. We also know $E = \frac{12 \cdot 4}{2} = 24$ because every face has $4$ edges and every edge is shared by $2$ faces. We can solve for the vertices based on this information.

Using the formula we can find: \[V + 12 - 24 = 2\] \[V = 14\] Let $t$ be the number of vertices with $3$ edges and $f$ be the number of vertices with $4$ edges. We know $t+f = 14$ from the question and $3t + 4f = 48$. The second equation is because the total number of points is $48$ because there are 12 rhombuses of $4$ vertices. Now, we just have to solve a system of equations. \[3t + 4f = 48\] \[3t + 3f = 42\] \[f = 6\] \[t = 8\] Our answer is simply just $t$, which is $\boxed{\textbf{(D) }8}$ ~musicalpenguin

Solution 5

Each of the twelve rhombi has two pairs of angles across from each other that must be congruent. If both pairs of angles occur at $4$-point intersections, we have a grid of squares. If both occur at $3$-point intersections, we would have a cube with six square faces. Therefore, two of the points must occur at a $3$-point intersection and two at a $4$-point intersection.

Since each $3$-point intersection has $3$ adjacent rhombuses, we know the number of $3$-point intersections must equal the number of $3$-point intersections per rhombus times the number of rhombuses over $3$. Since there are $12$ rhombuses and two $3$-point intersections per rhombus, this works out to be:

$\frac{2\cdot12}{3}$

Hence: $\boxed{\textbf{(D) }8}$ ~hollph27 ~Minor edits by FutureSphinx

Solution 6 (Based on previous knowledge)

Note that a rhombic dodecahedron is formed when a cube is turned inside out (as seen here), thus there are 6 4-vertices (corresponding to each face of the cube) and 8 3-vertices (corresponding to each corner of the cube). Thus the answer is $\boxed{\textbf{(D) }8}$

Solution 7 (Using Answer Choices)

Let $m$ be the number of $4$-edge vertices, and $n$ be the number of $3$-edge vertices. The total number of vertices is $m+n$. Now, we know that there are $4 \cdot 12 = 48$ vertices, but we have overcounted. We have overcounted $m$ vertices $3$ times and overcounted $n$ vertices $2$ times. Therefore, we subtract $3m$ and $2n$ from $48$ and set it equal to our original number of vertices. \[48 - 3m - 2n = m+n\] \[4m + 3n = 48\] From here, we reduce both sides modulo $4$. The $4m$ disappears, and the left hand side becomes $3n$. The right hand side is $0$, meaning that $3n$ must be divisible by $4$. Looking at the answer choices, this is only possible for $n = \boxed{8}$.

-DEVSAXENA

(Isn't this the same as the last half of Solution 2?)

Solution 8 (Dual)

Note that a rhombic dodecahedron is the dual of a cuboctahedron. A cuboctahedron has $8$ triangular faces, which correspond to $\boxed{\textbf{(D) }8}$ vertices on a rhombic dodecahedron that have $3$ edges.

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/GP-DYudh5qU?si=fFif-OiVZnkdTTv0&t=5105

~Math-X

Video Solution

https://youtu.be/5OuzPFvJPEY

Video Solution

https://www.youtube.com/watch?v=Z-OCnHUwnj0

Video Solution by OmegaLearn

https://youtu.be/0AG5XmWY-D8

Video Solution by TheBeautyofMath

https://www.youtube.com/watch?v=zvKijDeiYUs

Video Solution

https://youtu.be/0ssjr8KjOzk

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

See Also Cheese

2023 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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All AMC 10 Problems and Solutions

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