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

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Number 12. Vera built a tower of blocks. She wants to replace the two blocks with the question marks with two blocks with numbers.  She wants the number on each block in her tower to be at least two more than the number on the block below it. How many ways can Vera do this?  Feel free to pause if you want to try and solve it.  Alright, so what we can do here, we can see actually from the answer options that there's not going to be a ton of different ways to do this.  The largest number here is 7. So we can actually just try to count all the different ways she can do this. So let's start to do this in a sort of ordered way.  Let's start by calling this number, the one in the first question mark, 8. Because it can't be 7 because each one has to be at least two more. So the lowest number it can be is 8.  So if it's 8, then what can the second question mark be? Well, it could be 10 because it has to be at least two more. It could be 10, 11, 12, but it can't be 13, right?  Because if the second one was 13, it would be one less than 14, which is not allowed, right? It needs to be at least two less. So there's three options if it's 8, right?  Alright, what if it's the next number, 9? Well, if this first block is 9, then the next one has to be at least two more than that. So it could be 11 or 12, right?  Because it can't be 13 or higher as we already saw because it has to be at least two less than 14. So that leaves 11 and 12.  That's two options, and you can probably see where this is going. If we have 10 over here, then the only option for this block is for it to be 12, which is one option.  And then after that, if this is, for example, 11, there aren't enough options, right? Because the difference between 14 and 11 is 3.  So there's no way to put a block that's two more than 11 and also two less than 14. So that leaves 3 plus 2 plus 1 options, which is 6, and that is option D.

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