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Alexander, from Wilson's School, wrote:
Phoebe first drew one “down,” and then as many “inverted Cs” as needed to get seven squares. Therefore:
Alice first drew seven “alongs” at the top, and then seven “alongs” at the bottom. She then used eight “downs” to connect the gaps in between two parallel “alongs.” Therefore:
Luke first drew one “square,” and then six “inverted Cs” to make seven squares. Therefore:
Great! Laeticia, from Woodbridge High School, also gave correct general formulas here. She went on to comment on one of our later puzzles:
If the height is h and the width is w, then the perimeter is 2h + 2w. Then there are (h+1)(w+1) dots, and w(h+1) + h(w+1) lines.
Niharika solved the rest of our questions:
Each L-shape has an 'inner' L-shaped line and an 'outer' L-shaped line. If an L-shape has height and width n, then the outer L has length 2n and the inner L has length 2(n-1). There are 2 lines remaining on the ends of the L, so altogether the perimeter is 4n.
There are 2(n-1) lines left inside the L-shape, so the shape is made of 6n - 2 lines.
The number of squares in an L-shape is 2n-1: think of each L as two rectangles of height 1 and width n that overlap in one square.
Think of these as two separate overlapping squares, each with $n^2$ dots. They overlap in one dot, so in total there are $2n^2 - 1$ dots, and so $2n^2 - n - 1$ white dots.
In each square there are 2n(n-1) lines, and the lines never overlap, so in total there are 4n(n-1) lines.
Square of squares:
We can split a pattern like this of side length n up into four rectangles of height 1 and length n-1, so there are 4(n-1) edge squares.
There are 4n lines making up the outer square and 4(n-2) lines making up the inner square. There are 4(n-1) lines left in the middle. This gives 12(n-1) lines in total.
Dots and more dots:
Inside the squares there are $n^2$ dots, and on the vertices there are $(n+1)^2$ dots, so in total there are $2n^2 + 2n + 1$ dots.
In each row there are n lines, and there are n+1 rows, so the rows contribute n(n+1) lines. Similarly the columns contribute n(n+1) lines, so together there are 2n(n+1) lines.
Rectangle of dots:
There are 4n horizontal lines and 3n vertical lines, so 7n lines in total.
On each row there are 2n+1 dots, and there are n+1 rows, so there are (2n+1)(n+1) dots.
Fantastic! Thank you all.