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Fit These Shapes

Age 5 to 11 Challenge Level:

Many pupils seemed to have taken an initial logical step and decided that you can put $16$ of each shape into the square frame.This is true if the shapes are put into the smallest square they can fit in and then fit $16$ of these small squares into the large square frame. 

Some of these solutions were sent in by Deanna from Merseyside Primary School, Cerys and Leah from Lees Hill Primary Schools, Vachini from Riverside School and Simran from Brookfield School.

But other pupils obviously thought differently and went about it in different ways. Elizabeth from Cape Cod Academy in US found she could use $28$ triangles. Alex from Lees Hill Primary school found a way of using $24$ triangles. Just as I was putting these comments altogether we received this from Jemma in Island School (I think in Hong Kong):

When we look at the shapes the width and length from the top to the bottom is the same as the square e.g. the Diameter of the circle is the same as the lengths of the square. So no more circles than the amount of squares ($16$) can fit in to the frame. I believe this is the same for all the shapes no matter how much they are rotated. However the triangle can fit more along as we can fit more triangles upside down in the spaces in between the upright triangles.  

Heer, Heledd, Sarah, Nirali from St Stephen's Carramar, Australia sent in this solution.

$16$ squares, $16$ circles, $9$ octogons, $20$ pentagons and $35$ triangles.  

Thank you all for sending these solutions in during the summer, perhaps when you were on holiday! Others may like to take a second look and decide on a sure way of getting all the solutions.