|
|
| (62 intermediate revisions by 28 users not shown) |
| Line 1: |
Line 1: |
| − | ==Problem==
| + | Explore Google, the most popular and widely used search engine globally, known for delivering fast, relevant, and reliable search results. In addition to search capabilities, Google offers a wide array of services and products, including Gmail for email, Google Maps for navigation, Google Drive for cloud storage, YouTube for videos, Google Photos for image management, and Google Docs for collaborative document editing. With a commitment to innovation, Google continuously improves its technologies to enhance your online experience across all devices, providing seamless integration with Android, Chrome, and more. |
| − | 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>
| + | Website: https://www.google.com/ |
| | | | |
| − | ==Solution 1==
| + | Gmail: alaccary446@gmail.com |
| | | | |
| − | 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>.
| + | Hashtag: #Google trends #google news #google map #google drive |
| − | ~aiden22gao ~zgahzlkw (LaTeX)
| |
| − | | |
| − | ==Solution 2==
| |
| − | | |
| − | 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>A</math> be the number of edges with 3 vertices and <math>B</math> be the number of edges with 4 vertices.
| |
| − | 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>.
| |
| − | 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.
| |
| − | | |
| − | ~Technodoggo ~zgahzlkw (small edits)
| |
| − | | |
| − | ==Solution 3==
| |
| − | 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.
| |
| − | | |
| − | Using the formula we can find:
| |
| − | <cmath>V + 12 - 24 = 2</cmath>
| |
| − | <cmath>V = 14</cmath>
| |
| − | Let <math>t</math> be the number of vertices with <math>3</math> edges and <math>f</math> be the number of vertices with <math>4</math> edges. We know <math>t+f = 14</math> from the question and <math>3t + 4f = 48</math>. The second equation is because the total number of points is <math>48</math> because there are 12 rhombuses of <math>4</math> vertices.
| |
| − | Now, we just have to solve a system of equations.
| |
| − | <cmath>3t + 4f = 48</cmath>
| |
| − | <cmath>3t + 3f = 42</cmath>
| |
| − | <cmath>f = 6</cmath>
| |
| − | <cmath>t = 8</cmath>
| |
| − | Our answer is simply just <math>t</math>, which is <math>\fbox{(D) 8}</math>
| |
| − | ~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 <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 <math>2</math> <math>3</math>-point intersections per rhombus, this works out to be:
| |
| − | \dfrac{<math>2*12</math>}{<math>3</math>}
| |
| − | Which works out to be <math>\fbox{(D) 8}</math>
| |
Explore Google, the most popular and widely used search engine globally, known for delivering fast, relevant, and reliable search results. In addition to search capabilities, Google offers a wide array of services and products, including Gmail for email, Google Maps for navigation, Google Drive for cloud storage, YouTube for videos, Google Photos for image management, and Google Docs for collaborative document editing. With a commitment to innovation, Google continuously improves its technologies to enhance your online experience across all devices, providing seamless integration with Android, Chrome, and more.
Website: https://www.google.com/
Gmail: alaccary446@gmail.com
Hashtag: #Google trends #google news #google map #google drive
