Elliott from Wilson's School described what Phoebe, Alice and Luke did:

Phoebe began with one vertical match stick and then added seven inverted C shapes of match sticks to make seven squares. Overall, there were seven inverted Cs and one downs, making 22 all together.

If there were 25 squares, there would be 1 downs and 25 inverted Cs. There would be 76 match sticks in total.

If there were 100 squares, there would be 301 match sticks and if there were 1000001 squares, there would be 3000004 match sticks.

You can work this out by multiplying by 3 and adding 1.

Alice made the seven squares by placing the horizontal match sticks down first, then placing the vertical matches to join them up.

There were 14 'alongs' and 8 'downs', totalling 22 match sticks.

If there were 100 squares, there would be 301 matches and if there were one million and one squares, there would be three million and four match sticks.

You can find this by doubling the number of squares and adding the number of squares plus one.

Luke made seven squares by making the first square, and then adding inverted Cs. There was one square and six inverted Cs in total.

If there were 25 squares, Luke would make one square and then add 24 inverted Cs, using 76 match sticks altogether.

If there were 100 squares, there would be 301, and if there were one million and one squares, there would be three million and 4 match sticks.

You can work this out by subtracting one from the number of squares, multiplying by three and adding four.

Leonie and Pippa from the Mount School in York described what Phoebe and Alice did and then pictured what would happen when the number of squares increased:

Phoebe started with 1 vertical matchstick. She added 3 more matchsticks at a time to make a square.

Alice started with laying out all the top horizontal row of matchsticks, then she added the bottom horizontal row of matchsticks. She then added the vertical matchsticks.

For 7 squares, there are 8 downs and 14 alongs. There are 22 matchsticks in total.

For 25 squares, there are 26 downs and 50 alongs. There are 76 matchsticks in total.

For 100 squares, there are 301 matchsticks in total.

For 1 million and 1 squares, there are 3000004, matchsticks in total.

Sophie and Rachael, also from the Mount School in York, included a formula:

Where N is the number of squares.

Alice laid out all the top matchsticks then all the bottom ones, then all the middle ones.

Alongs = 2N

Downs = N+1

Altogether = 3N+1
 
 
Jamie, from Highfields School in Derbyshire, sent us this clear solution to the L shapes extension problem.
 

Hannah from Fullbrook had a go at a couple of the extension questions:

Growing Rectangles:

For a rectangle with a height of two and a width of 25, the number of dots required would be $(2+1) (25+1) = 3 \times26= 78$.
Therefore, 78 dots are required.
The number of lines required would be $76 + (2\times25) + 1 = 76 + 50 + 1=127$
Therefore, 127 lines are required. (The number 76 is known from the Seven Squares problem.)

For a rectangle with a height of two and a width of 100, the number of dots required would be $(2+1)\times(100+1) = 3 \times101 = 303$.
Therefore, 303 dots are required.
The number of lines required would be
$301 + (2\times100) + 1 = 301 + 200 + 1=502$.
Therefore, 502 lines are required.

For a rectangle with a height of 2 and a width of n, the number of dots required would be $3(n+1)$, and the number of lines required would be $5n + 2$ (because $(3n+1)+(2n+1)=5n +2$)

L shapes:

If it is of height 25, width 25.
               Perimeter: 100
               Number of squares: 49
               Number of lines: 148
             
If it is of height 100, width 100
              Perimeter: 400
              Number of squares: 199
              Number of lines: 598

If it is of height n , width n.
              Perimeter: 4n
              Number of squares: 2n-1
              Number of lines: 6n -2  
 
Laura from Ramapo submitted a clearly explained solution:
Pattern with a height of 2 and a width of 25:
Perimeter (add total number on each side) $= 2 + 2 + 25 + 25 = 54$
Number of dots: 26 dots on each line, total of 3 lines $26 \times 3 = 78$
Total number of lines: $54$(perimeter) $+ 25$ (middle line) $+ (2 \times24) =127$.  $2 \times 24$ represents the vertical lines within the rectangle.
 
Pattern with a height of 2 and a width of 100:
Perimeter (add total number on each side) = $2 + 2 + 100 + 100 = 204$
Number of dots: 101 dots on each line, total of 3 lines $101 \times 3 = 303$
Total number of lines: $204$ (perimeter) $+ 100$ (middle line) $+ (2 \times 99) = 500$
$2 \times 99$ represents the vertical lines within the rectangle.
 

 
Elijah, who is home-educated, sent us a comprehensive answer to the entire problem. Bravo!
GROWING RECTANGLES
height $2$, width $n$
$P = 2(n+2)$
dots $= 3(n+1)$ [there is one more dot than line on each side]
lines $= 3n$ [the vertical lines] $+ 2(n+1)$ [the horizontal lines] $= 5n + 2$
height $m$, width $n$
$P = 2(m+n)$
dots $= (m+1)(n+1)$
lines $= (m+1)n + m(n+1) = 2mn + m + n$
 
L SHAPES
height $n$, width $n$
$P = 4n$
squares $= 2n-1$
lines $= 4n + 2n-2 = 6n-2$

height $m$, width $n$
$P = 2(m+n)$
squares $= m + n - 1$
lines $= 2m + 2n + m + n - 2 = 3m + 3n - 2$
 
TWO SQUARES
4 black
white = 27
lines = 48

25 black
white = 1224
lines = $25\times24\times4 = 2400$

100 black
white = 10000 + 10000 - 101 = 19899
lines = $100\times99\times4 = 39600$
  
$n$ black
white = $n^2 + n^2 - (n+1) = 2n^2 - n - 1 $
lines = $4n(n-1)$
 
SQUARE OF SQUARES
side length 5
edge squares = 16
lines = 48
 
side length 25
edge squares = 96
lines = $25\times4 + 24\times4 + 23\times4 = 288$

side length 100
edge squares = 396
lines = $100\times4 + 99\times4 + 98\times4 = 1188$

side length $n$
edge squares = $4n - 4 = 4(n-1)$
lines = $4n + 4(n-1) + 4(n-2) = 12n - 12 = 12(n-1)$

height $m$, width $n$
edge squares = $2m + 2n - 4 = 2(m+n-2)$
lines = $2m + 2n + 2(m-1) + 2(n-1) + 2(m-2) + 2(n-2) = 6m + 6n - 12 = 6(m+n-2)$
 
DOTS AND MORE DOTS
side length 3
dots = 25
lines = 24
 
side length 25
dots = $625 + 676 = 1301$
lines = $25\times26\times2 = 1300$

side length 100
dots = $10000 + 10201 = 20201$
lines = $100\times101\times2 = 20200$
 
side length $n$
dots = $n^2 + (n+1)^2 = n^2 + n^2 + n + n + 1 = 2n^2 + 2n + 1$
lines = $2n(n+1) = 2n^2 + 2n$
 
height $m$, width $n$
dots $= mn + (m+1)(n+1) = mn + m(n+1) + (n+1) = mn + mn + m + n + 1 = 2mn + m + n + 1$
lines $= 2mn + m + n$
 
RECTANGLE OF DOTS
For 2 squares:
side length 3
lines = $7\times3 = 21$
dots = $4\times7 = 28$

side length 25
lines = $7\times25 = 175 $
dots = $26\times51 = 1326$

side length 100
lines = 700
dots = $101\times201 = 20301$

side length $n$ 
lines $= 7n$
dots $= (n+1)(2n+1)$
 
Generalising to $p$ squares of side length $n$
lines $= (3p+1)n$ [got $3p+1$ from Tom's matchsticks]
dots $= n + 1 + pn(n+1)$ [start with n+1 dots, then for every square added, need to add $n(n+1)$ dots] $= (n+1)(pn+1)$