Question number 13. Masuda and his friends built the number 2022 using 66 cubes as shown in the picture. They painted the whole surface of the structure yellow. How many of the cubes have exactly four faces painted? So the first thing to note is that any cube that is connected to exactly two other cubes has exactly four faces painted. If it's connected to exactly one other cube, then it has five faces painted. And the first thing, or the second thing, I guess, that we can see over here is that pretty much none of these have anything more than that. None of these cubes are on their own. So none of them, they all have at least one connection to another cube. And none of them have three faces connected to any other. None of the cubes have three faces connected to other cubes, which we can kind of tell because this is only in two dimensions over here. So now getting to the digits, there are only two types of digits here. So there's only a two and there's a zero. And the digit zero consists only of cubes with exactly four faces painted because this is a loop. Every single one has two cubes that are connected to it. The corners have these two edges. Each edge has the one on either side of it. And so every single cube in this zero has four faces painted. But what about the digit two? Well, in the digit two, there are two cubes, one at each end that has five faces painted because they're only attached to one other cube. So there's going to be a total of two times three cubes that do not have four faces painted. They have five faces painted. So one, two, three, four, five, six, if you're following my mouse, hopefully. So six of these 66 cubes have five faces painted. And since we know that all the cubes either have four faces or five faces painted, we can say that 66 total cubes minus six cubes with five faces painted gives us 60 cubes with four faces painted. And that answer is E.

cubes

painted faces

structure

digits

2022