We had many excellent solutions sent in for this challenge. I mention below all those who explained their thinking. These first two were particularly good and worthy of being included in full.

Lucy from Hertsmere Jewish Primary School in England wrote:

I worked out the purple square had to be two because three times three equals nine and nine times three equals twenty-seven which you can't do because the number has to be less than twelve and you can't do one because no two shapes can be the same number but two times two is four and four times two is eight.
Then I filled out the yellow semi circles as eight and the purple squares as two.
Then I worked out the second one down using what I already knew. Two times four is eight so I filled out all the orange ovals as four.
Then I worked out the red triangle equaled zero because only zero can times by a number and equal zero. Then I filled out all of the red triangles as zero.
Next I worked out the blue rectangles were three because one times one equals one which you can't do; I'd used two and four times four is sixteen which you also can't do so three times three is nine so the green star is nine. Then I filled out all the blue rectangles as three.
Then I worked out three times two equals six so I filled out all the green triangles as six.
Next I worked out six times two is twelve so I filled out all the red circles as twelve.
After I worked out three times one is three so the yellow diamond is one. Next I filled out all the yellow diamonds as one.
Next I looked at yellow diamond (one) times blue hexagon equals blue hexagon, and purple square (two) times purple star equals blue hexagon and thought nothing times two can equal seven so it had to be ten.
Finally I filled out all the blue hexagons as ten and the purple star as five seeing as two times five is ten.

Then, from Jenny, PhiPhi, SeoHyun, Ly from the Britsh International School in Vietnam, we had:

The square represents: 2, because 3x3x3 is 27 which is over 12 and if you do 1x1x1 it will equal 1 and the answer is not the 1.
The semi circle represents: 8, because 2x2x2 is 8.
The rectangle represents: 3, because 3x3 is 9 and it can't be 2x2 because the square is already 2. It can't be 4x4 because the answer is over 12 and it can't be 1x1 because the answer would be the same.
The oval represents: 4, because the square is 2 and the answer is 8 so that means it has to be 4 because 2x4 is 8.
The upside down isoceles triangle represents: 0, because the square is 2 and 2x0 is 0 and the semi circle is 8 and 8x0 is 0.
The triangle represents: 6, because in one of the equations were 3x2 and 3x2 is 6.
The hexagon represents: 10 because in the equation it says 1x10 and 1x10 is 10.
The eight-pointed star represents: 9, because 3x3 is 9.
The diamond represents: 1, because in the other equation, it says 1x10 and the answer is 10.
The five-pointed star represents: 5, because in one of the equations 2x_ = 10, ($10\div2=$_) so the answer must be 5.
The circle represents: 12, because in one of the equations there was 6x2=_, so the answer must be 12.

These are both exceptionally good explanation of their thinking. Well done!

We also had solutions from the following:

Grace, Harry and Leo from Alderman Jacobs School, England.
Reuben, Izzy and Bella, Myah and Katy, Katy (I don't know whether this is the same Katy!) from Mawnan Primary School, Falmouth.
Tsubasa, Filippo and Guglielmo from the International School of Turin, Italy.
Lois from Greenacre School for Girls in Surrey.
P7 from Letham Primary School in Angus, Scotland.

Finally, Jad, Anay, Nishk Zihan from JPS in Dubai UAE sent in this picture of their work:

Well done all of you.  These were fine examples for others to learn from.