is problem 29 of the grade 11 and 12 2025 contest. Patricia has written a number in each cell of a 7x10 table. The sum of all the numbers in any 3x4 or 4x3 rectangle is 0. The numbers in two of the cells are shown in the diagram. What is the sum of all the numbers in the table? Okay, our answer choices are a, negative 5, b, negative 20, c, negative 25, d, negative 45, and e, it is not possible to determine. So maybe there are like two valid constructions that lead to different sums. Well, okay. In my mind, we could try like two things. So the first approach is to like try to make our own pattern. Our own pattern. And that, well that would be, that would require filling out 70 things. And that might kind of be difficult. So I vote, maybe we don't try this. Maybe you can try this like by yourself and see if you can come up with like a general pattern, like a general maybe like formula for doing it. But maybe there's a different way of doing it. And a different way of doing it, in my mind, is to carve the shape up into regions, into regions of zero. So into zero sum regions. So we know that the sum of any numbers in a 3x4 or 4x3 rectangle is 0. So if I were to draw some like arbitrary 4x3 rectangle here, we know that the sum of all of these things is 0, which is nice. We like that. But if we're maybe a little more clever on how we pick these things, we can achieve something pretty cool. So I'm going to make the following cuts in our shape. And we're going to have a little bit of overlap, but we can deal with that later. Okay. So with these cuts, we know that, and I'm going to color some stuff in, we know that this is 0, this is 0, this is 0, and this is 0. Okay, awesome. We've just cut out a bunch of the rectangle. And then something we also know is because these like two corner bits, or like, yeah, these center bits add to 45. Then what we know is that this thing in here, like these two values, it's going to be negative 45. And this one will also be negative 45, because they need to cancel out the 45 that exists in both of them. So if we add all of these sums together, we can add 0 plus 0 plus 0 plus 0 for our four, you know, yellowed out 0 rectangles, minus 45, minus 45 for our irregularly shaped, like non rectangles. And then finally, like the remaining 45 from the center. And if you sort of look at it, we've accounted for the entire thing. And this sums to exactly negative 45. So our answer should be negative 45. We didn't make any assumptions. So this is perfectly valid. And I think you should try to tessellate the plane with numbers, such that this actually works for yourself, because it's a very fun puzzle. And as a little bit of extension, like tiling the plane like this is actually a sort of difficult problem in general. And some very interesting problems can be made out of it.

sum of numbers

7x10 table

zero-sum patterns

tiling problems

problem-solving approach