We received many correct solutions to this puzzle:
Josh from Swalcliffe Park School sent us this solution:
Elizabeth, Holly, Kelly, Harrison, Harry and Matthew from Miss Rowcliffe's Maths Class in Upton Heath C of E Primary School in Chester sent us this solution:
Numbers less than 20  5  18  15  11  1 
Numbers more than 20  30  36  45  23  25 
Odd numbers  35  21  55  7  9 
Even numbers  20  24  10  2  16 
Factors of 60  60  12  6  3  4 
Multiples
of 5

Multiples
of 3

Triangular
numbers

Prime
numbers

Square
numbers

Lydia from Woolmer Hill School sent in this solution:
Numbers lesss than 20  16  1  18  15  7 
Numbers more than 20  25  30  60  35  23 
Odd numbers  9  5  45  55  11 
Even numbers  4  12  24  20  2 
Triangular numbers  36  6  21  10  3 
Square
numbers 
Factors
of 60

Multiples
of 3

Multiples
of 5

Prime
numbers 
Vicky and Katrina, also from Woolmer Hill School, submitted this solution:
Numbers lesss than 20  4  18  6  10  11 
Triangular numbers  1  21  15  45  3 
Odd numbers  25  9  5  55  7 
Even numbers  16  12  30  20  2 
Numbers more 20  36  24  60  35  23 
Square
numbers

Multiples
of 3

Factors
of 60

Multiples
of 5

Prime
numbers

Ben, Henry, Joseph, Simon, Michael and Jake from Hampton School sent us these three different solutions.
Heidi, Lucy and Natasha from Millais School sent us this record of their work:
Initially we looked at the headings and found those that could not have any overlap,
e.g. Odd and evens, primes and square numbers, numbers under and over 20.
We placed those headings first and then put the others in place.
Here is their completed board.
Poppy and Hazel, also from Millais School, reported that:
We started by trial and error and got rather frustrated!
We realised that some of the headings had to be paired up because number could not appear in both categories, e.g. odds and evens, primes and squares.
From there we placed the headings in the table and gradually moved the others around until it worked. Interestingly, our solution was different to others in our class.
Lara and Jeni, again from Millais School, sent us a different solution
Oliver from Swalcliffe Park School in the UK sent in this solution:
Congratulations also to Sam from Ridgewood School, Amy from SMTS and Polly and Heather from Hertfordshire and Essex High School who all sent us correct solutions. Well done to you all.