# Four squares into many squares

You are given four unit squares and your task is to form as many rectangles as possible out of it starting from 1 square (by overlapping every squares into each other) to N, one by one (2,3,4...). So you start with 4 unit squares overlapped each other (which will be counted as one rectangle).

But in order to do that, you are allowed to move only 1 square at a time and you need to count one more rectangle than previous turn. That means let's say you have counted 5 rectangles previously, on the next turn, by moving any but one square, you are supposed to find exactly 6 rectangles. If it is not possible, that would be your answer, not necessarily the maximum number of rectangles possible.

At most how many rectangles can you form by moving only one square as you wish?

and

At most how many rectangles can you form starting from 1 rectangle if you are allowed to move as many squares as possible every turn?

If this question was asked for 2 squares, the answer would be 3 as shown below; • I don't understand the problem at all. It asks two questions but the sample only gives one answer. I am confused about what you consider to be a rectangle from this sentence: So you start with 4 unit squares overlapped each other (which will be counted as one rectangle). And what does "moving a square" mean? Is there a difference between a square and a rectangle, in this problem? – Weather Vane Feb 6 '19 at 10:18
• @Nautilus show me as a picture? if you count more than 4, it doesnt satisfy. – Oray Feb 6 '19 at 11:12
• @WeatherVane sorry for confusion, if you overlap all squares, you will able to count 1 square or rectangle right? it has to be the start point since there is no other way aronud to have 1 rectangle. then if you move out one of the square inside on that overlapped square as show above, you would have 2 squares or rectangles and you would have moved only one square at a time... and so on. and a square is actually a rectangle... – Oray Feb 6 '19 at 11:14
• I was going to post one, but realized reaching 4 is actually impossible, so it is 3. Then again, making 6 rectangles from 2 squares is possible by overlapping an edge. – Nautilus Feb 6 '19 at 11:28
• @Nautilus yes it is possible but since 4 is not impossible, the answer becomes 3. – Oray Feb 6 '19 at 14:08

My best (so far)

Up to 12 rectangles, moving one at a time. Maximum 36 rectangles. These are getting really hard to count, so I'm stopping at what I think might be a 24. Any double-checking help would be much appreciated.

• looks like at least the 21 is wrong – Bass Feb 7 '19 at 3:41

Ok I spent way too long on this, so I got to 21 and then included some bonuses, including 36, 38, 40, 42, 44; some important shapes; and an estimated maximum shape. I don't have any more time but I suspect you could spend many days on this. Enjoy! p.s. I did this in a dark theme :)

p.s. After 10 there may be some I missed, feel free to comment any corrections. I think 18 might really be 19 btw. 19 I didn't actually count but subtracted and then added the new ones, may be right, but can't be 100% sure without spending a lot more time on it;

EACH one of these past 10 is really a puzzle in itself ["how many maximum rectangles?"], and with each consecutive one it gets harder and hader to count.

Past #21, for ease of counting will require resizing it up larger so you can more easily count the tinier boxes.

Have fun and enjoy! Especially the potential maximum, wow.

This is a great puzzle and it is possible that people could even spend years on this, especially since with each consecutive one it gets exponentially harder to be sure you counted all the boxes ;)

NOTE: A series of puzzles for EACH set (11, 12, 13, etc) would be definitely good as their own question - in fact, all the way up to whatever is the possible maximum. • The middle important piece should be a 15 though: grouping by the top left corner, there must be 5+4+3+2+1 rectangles. – Bass Feb 7 '19 at 3:26