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Isabelle from South Wilts and Natalie from St.Andrews International School in Thailand both answered the first challenge correctly.

Here is Natalie's arrangement of the shapes:

 

Here is how Isabelle described her strategy:

a) Write down the areas and perimeters of each shape.

b) The only three shapes that share an area are G, A & C therefore they must occupy the middle column.

c) B, D & I have areas less than 14 so must occupy the left column and E, F & H have areas greater than 14 so must occupy the right column.

d) The shapes with perimeter 20 (B, C & H) must go in the bottom row, those with perimeter 18 must go in the middle row and those with perimeter 16 must go in the top row.

There is only one way this can be achieved so by a process of elimination the solution is as above.

 

 

Isabelle also answered the second challenge correctly:

 

 
      AREA  
    - = +
  - 2 by 7 4 by 4 3 by 6
PERIMETER = 1 by 9 2 by 8 5 by 5
  + 1 by 15 1 by 16
3 by 8
 
Krystof from Uhelny Trh, Prague, filled in one of the squares of the extended grid:
 

It's possible to fill in the box on the left too. I wonder if you can think of a way, and convince yourself that it's impossible to fill in the top and right boxes.