Grades 1-2 Video Solutions 2020-2020Levels12prob23

Video Transcription

Problem number 23. Six different numbers chosen from 1 to 9 are written on the faces of the cube,  one number on each face. The sums of numbers on each pair of opposite faces are equal.  Which number could be on the face opposite the face of the number 5? A3, B5, C6, B7, E9. Since this problem asks which of the numbers listed can be used opposite the face of the 5,  a valid way to solve this problem is to try each one of these numbers and see which ones cannot work and which one can.  We need to look at the sum that is made of the number given in the answer and 5.  5 plus 3 is 8. We would need to make the other sums also 8, but this is impossible.  We already have 8 on one of the faces and we cannot use the digits 0 on the opposite face because we only use the numbers from 1 to 9, so there's no way to make that 8. Also,  we need to put another 4 opposite the 4 and we can't use the number 4 because we're using 6 different numbers. So, 3 will not work.  B5? Well, 5 is already used, so we cannot use that again. That also doesn't work.  C6. If we put 6 opposite the 5, the sum is 5 plus 6 is 11. Opposite the 4, we would have to put a 7 because 4 plus 7 is 11. And opposite the 8,  we would need to put 3, which is 11. So, the 6 numbers that would be used  and written on this cube would be 5, 6, 4, 7, 8, and 3, which seems to work.  Let's look at the remaining choices. 7. 5 plus 7 is 12.  We need to add 4 to the 8 to make 12, but we already see that the 4 is not opposite the 8, so this will not work. And then 9. 5 plus 9 is 14.  Since we have a 4 on one of the faces, we would need to put 10 opposite it,  but 10 is not a number that we can use. We can only use numbers from 1 to 9, so it also does not work. So, the correct answer is C.  6 is a number that can be on the face opposite the number 5.

Video Summary

In this problem, numbers from 1 to 9 are placed on a cube's faces, with sums of numbers on opposite faces being equal. To find which number can be opposite the number 5, candidate numbers were tested. After examining different sums, it was determined that if 6 is opposite 5, the other sums are valid: opposite 4 is 7, and opposite 8 is 3, forming correct pairs with a sum of 11 (5+6, 4+7, 8+3). Other options failed to meet criteria due to repetition or exceeding number availability. Thus, 6 is correctly positioned opposite 5.