Grades 5-6 Video Solutions 2025-2025_5-6

Video Transcription

Number 19. Henry wants to write a 0 or a 1 in each cell of the diagram, so that the sum of the numbers in each row, column, and diagonal is 3.  He has already written a 0 in one of the cells. When he finishes, what will the sum of the numbers in the cells with the question marks be? Pause if you want to try and solve it.  The first thing to notice here is that since he's writing either a 0 or a 1, and he wants the sum of the numbers in each row, column, or diagonal to be 3,  he needs to have 3 1's in each row, column, or diagonal. He needs to have 3 because that's the only way that he can get a sum of 3.  The rest of the numbers are exactly one other number because there's 4 total, 1, 2, 3, 4 cells. The other number has to be a 0 because 3 1's plus 1 0 is equal to 3.  Notice in this cell over here, he's written a 0. For example, for this column, he already has the 1 0, which means that every other number in this column has to be a 1.  This is a 1, this is a 1, and this is a 1. The same is true for this row. He already has the 1 0, so all of the other 3 have to be 1's.  The same is true for this diagonal over here. This is a 1, this is a 1, and this is a 1 because he already has that 0 there.  But how does this help us get the numbers with the question marks? What we can notice is that he already has 2 1's in this row.  If he already has 2 1's in this row, then that means that one of these numbers has to be a 0 and the other one has to be a 1.  There's no other possibilities. If they're both 1's, then he would have 4 1's, and that's not a sum of 3.  If they were both 0, then that would be a sum of 2. 0 plus 1 plus 1 plus 0 is 2, and that's also not 3.  One of these is a 1 and one of these is a 0. We don't know which one, but it doesn't matter because the sum of a 1 and a 0 is always 1.  The sum of this question mark and this question mark, we don't know which is which, but their sum is 1.  The same is true for these two. We have 2 1's here, so one of these question marks is a 1 and one of these question marks is a 0.  We don't know which is which, but we know that their sum has to be 1 because 1 plus 0 is 1.  Now if we add these two 1's, 1 plus 1 is 2, which is option B, so the correct answer for number 19 is B.

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