Grades 5-6 Video Solutions 2024-2024_5-6

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Question 26. Mary wants to write the numbers 1 to 8 on the vertices of a cube. She wants the sum of the numbers of the vertices of each face to be the same. She has already  written the numbers 6, 7, and 8, as shown. What number should she write on the vertex marked with a question mark? The sum of all the vertices, which use the numbers 1 through  8 exactly once, is 36. We can observe that when we consider the 4 vertices of any face, the remaining 4 vertices form another face. Since the sum of the vertices on each face  is the same, and the total sum of all 8 vertices is 36, the vertices of each face must add up to half of 36, which is 18. Now, let's look at individual faces, more specifically  the face that has the 7 and the 8. The sum of the known vertices is 15. For the total to reach 18, we need the other 2 vertices to add up to 3. There's only one way to do  this with the numbers 1 through 8, and that is to use 1 and 2. Now, let's look at the bottom face with the 6, 8, and question mark. The sum of the known vertices is 14. For the  total to reach 18, we need the other 2 vertices to add up to 4. There are two ways to do this with numbers 8 or less, and they are 1 and 3, or 2 and 2. However, since we can  only use each number once, 2 and 2 does not work, leaving us with only 1 and 3. Since both faces need a vertex labeled 1, they must share it, which means the vertex under the  7 must have a 1. Therefore, the remaining vertex on the bottom face, the one with a question mark, must be a 3.

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