[Puzzler] Squaring Square Squares

[Mike]

(Stolen from 538's Puzzler series)

There are many ways to slice a big square into smaller squares (not necessarily of equal size), so that the smaller squares don’t overlap, while still making up the entire area of the big square.

For example, you can slice the big square into four smaller squares, each a quarter of the area of the big square. Or you could slice it into seven squares, if you take one of those four squares and slice it into four yet smaller squares.


What whole numbers of squares can you NOT slice the big square into?

[Stan]

The smartass answer is 1. Once you've made a slice, you have more than one square.

You can't do 2 or 3 unless you allow extra cuts and then putting pieces back together.

There's probably a formula list all options but that's too much thinking for me. I'll leave it to Flameblade as a belated birthday present.

[Mike]

I have an answer, and I'm sure it's right, but I haven't conclusively proved that the numbers I name are impossible. I've proven that all the other numbers ARE possible, and I instinctively know that the remaining ones aren't.

And yeah, 2 and 3 are on my list of impossible numbers. There aren't many others.

[FlameBlade]

I don't really understand what the puzzle is asking

[Mike]

I know you've already figured out what it means, but to clarify for others:

You have a square. You are subdividing the square by drawing lines inside of it. When you are done drawing lines, the original square will be subdivided into some number of smaller squares. These squares will fill the entire area of the original and will not overlap. Smaller squares do not have to be the same size.

[FlameBlade]

I re-read. I still have no idea what the puzzler is asking.

It seems like each time you divide a square, you lose the original square, and gain 4 squares. So you have a sequence that is basically add 3 squares, no?

[FlameBlade]

Oh. I see the problem. I could get 9 squares. Yeah, that question needed more examples.

[FlameBlade]

Solution with explanation. I hated how this puzzle was written because it did two fourths in a row. Should have done one fourth then one ninth. Took me longer to understand the question than to work it out. Which brings me to the point about communication is extremely important in math. Anyway, my solution below. Complete solution.

Okay.assuming 1 is legal.

There are operations of form n²-1. That is each time you divide a square, you lose a square, and add on n² squares, where n is 2 or greater.

So you have additive values of 3, 8, 15, 24 and so on.

We don't care about 15 and 24 because that's handled by 3.

So you start with 1, so you start to step out by 3. Start to sieve.

1, 4, 7, 10, 13, 16, 19, 22, 25,...
9, 12, 15, 18, 21, 24, 27, ...
17, 20, 23,...

Notice how all of the number of squares are possible after 16, because of slices in mod 3.

So, taking it all together, the impossible number of squares are...

2, 3, 5, 6, 8, 11, 14. Total of 7.

[Mike]

Close. Those values are also subtractive in some cases.

[FlameBlade]

Have absolutely no idea what you are talking about.

[Mike]

I'm saying that if you have nine squares in a 3x3 grid, then you could combine 4 of them into a larger square, leaving you with 6. So you had 9, but combining allowed you to subtract 3.

[FlameBlade]

Okay. That just means to me that the question is just badly written because there's nothing that really suggested that. Slice was never defined.

[Mike]

I don't know. It could have been better, but I just had to read it a couple times to sort it out. I think you locked in on the example and got some tunnel-vision. Happens to me all the time. I'm just happy to actually get one of these at all. I love doing them, but I'm generally not very good at them. You normally blow past me on stuff like this.

[FlameBlade]

See, not once in the question it says you could combine. By the header of the question, you get an impression that you only can generate square from squares by cutting it. In order to get to 6, you have to make rectangles somewhere in the process, so that was what caused me to rule it out.

[FlameBlade]

But I suppose you could just cut everything at once, which is another question.

[FlameBlade]

So it's just 2, 3, 5.

[Mike]

That is what I got, yes. We'll see next week how right that is, but I'm pretty sure.

[Mike]

But I suppose you could just cut everything at once, which is another question.

You have to cut everything at once. Only the finished product matters. Otherwise you can't cut anything at all. Cutting a big square into four smaller squares requires first cutting it into two rectangles, and then cutting those in half to make four squares. Worrying about the intermediate steps makes the whole thing impossible.

[FlameBlade]

Phew, If you ask me, the example is really really really terrible leading on. Removing the example seems to improve the puzzle significantly.