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2023 AMC 10A Problems/Problem 17: Difference between revisions

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A rhombic dodecahedron is a solid with <math>12</math> congruent rhombus faces. At every vertex, <math>3</math> or <math>4</math> edges meet, depending on the vertex. How many vertices have exactly <math>3</math> edges meet?
A rhombic dodecahedron is a solid with <math>12</math> congruent rhombus faces. At every vertex, <math>3</math> or <math>4</math> edges meet, depending on the vertex. How many vertices have exactly <math>3</math> edges meet?
<math>\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math>
==Solution==
==Solution==
We see there are <math>\frac{12 \cdot 4}{2}</math> edges. We have by Euler's Polyhedral Formula, <math>V-E+F=2</math> meaning <math>V-24+12=2</math> or <math>V=14</math>. Let there be <math>a</math> vertices that have <math>3</math> edges meeting and <math>b</math> vertices that have <math>4</math> edges meeting. Hence, <cmath>a+b=14</cmath>
We see there are <math>\frac{12 \cdot 4}{2}</math> edges. We have by Euler's Polyhedral Formula, <math>V-E+F=2</math> meaning <math>V-24+12=2</math> or <math>V=14</math>. Let there be <math>a</math> vertices that have <math>3</math> edges meeting and <math>b</math> vertices that have <math>4</math> edges meeting. Hence, <cmath>a+b=14</cmath>

Revision as of 19:58, 9 November 2023

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

We see there are $\frac{12 \cdot 4}{2}$ edges. We have by Euler's Polyhedral Formula, $V-E+F=2$ meaning $V-24+12=2$ or $V=14$. Let there be $a$ vertices that have $3$ edges meeting and $b$ vertices that have $4$ edges meeting. Hence, \[a+b=14\] \[3a+4b=48\] We find $b=6$ and $a=8$, hence the answer is $8$.

~SirAppel

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