Hello, this is problem 10 on the grade 11 and 12 2025 math kangaroo. In the xy plane in the region defined by zero less than or equal to x less than or equal to one, zero less than or equal to y less than or equal to one, i.e. a unit square with zero and one. Some points are painted black, the point x comma y is painted black if we're both x and y, the first digit after the decimal point is odd. What does the result look like? So we're left with all of these drawings, a I'll just let you observe them. We have rows and columns filled in a circle, a checkerboard diagonals, and then rows and columns maybe filled out. Okay, try it yourself. So yeah, we have all of these drawings. And while we could like go through and sample points, or maybe look for exclusions of things, um, it would help to maybe do this a little more systematically. So we can actually determine it rather than different from answer choices. So, very, roughly, we have this unit square here. And we have that the odds should be colored in. So like this column should be colored in. Then this column should be colored in. And so on. And then similarly, this row should be colored in this row. Or at least these are the places where each thing are like, each x and y are even and odd. So they're odd here. And they're odd here. But then we need to check when they're both even and odd. And when they're both even and odd, well, that's when these pillars overlap. So with like a slightly better graphic that we just want to work with over here, we get these two, like really nice sort of graphs on where x and y are odd. And then we just want to check where they overlap, because where they overlap is precisely where they like, have the and statement satisfied, then this just turns out being the intersection of these rows. So we're just left with a bunch of these little squares. And this looks like E.

Math Kangaroo

unit square

xy-plane

checkerboard pattern

odd digits