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I first took all the colours which were shown at the same time as the blue (because opposite blue would be on the bottom of the dice when blue is on the top). They are shown in the 'can't' list. Then that left orange when red was at the top. So orange was the colour opposite blue.
Dicey
Age 7 to 11
Challenge Level 
Sophie and Ruyi (Annesley College, Adelaide, South Australia) explained how they worked out their answer:
First we looked at the three pictures of the dice on your web
page, and memorised the order which the colours on the dice
took.
We then focused on the last two dice and noticed that the
orange (which was on the first dice) was not showing on the last
two dice.
Using our memory of the colours' relations to each other, we
realised that the orange must be opposite the blue.
Catherine (Tattingstone Primary UK) got the same answer, but thought it through a little differently:
I worked this out by making two lists. One list was of 'can't' and one of 'can'.I first took all the colours which were shown at the same time as the blue (because opposite blue would be on the bottom of the dice when blue is on the top). They are shown in the 'can't' list. Then that left orange when red was at the top. So orange was the colour opposite blue.
Ece (Private IRMAK Primary & Secondary School, Istanbul, Turkey) showed the position of each colour on the dice by drawing a net of a cube:
Jason (Priory School UK), Alison (UK), Thomas (Tattingstone School UK), Christina (Malborough Primary School, UK), Evren and Christian , Eda , and Ece D (Private IRMAK Primary & Secondary School, Istanbul, Turkey) all agreed that the colour was orange.
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