We had just a few solutions sent in from pupils trying to give some kind of proof for this challenge. Jasmine   from Salcombe Prep.  School in London sent in this picture and wrote:

Every time the numbers will make a rectangle

and from Michelle at the International School in the Seychelles we were sent the following:

When you multiply an even and an odd together it will always be an even number and when you times even and even togther it will also be an even number and how I did this was with cubes and I showed my teacher and he agreed with me.

Chris  from St. Mary's Catholic Primary,  England  had a good way of showing it:

An odd number multiplied by an even number will make an even number.
Key: different colour = different number

Finally from Christopher and Rei  at Seoul Foreign School (SFS) , South Korea (Republic of Korea) we have the following:

For example:
4*3=12  that's the same as 4+4+4
4 is even.
Everytime an even is added to a even the answer is always a even number.

We hope that we convinced you that an odd number times an even number is always an even number.

Year 2 at St. Matthews Primary School, Luton, England.  sent in during 2015 the following:

They also added a sentence or two saying that they know that Odd times Even will always be even.
Thank you very much

Just a note to say you can always email in to Nrich solutions to the probemns that are no longer "live".

Thank you for all the contributions.  It obviously made you think - and thinking is good!