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Thank you to everybody who sent in their solutions to this problem. Taylor said:

I chose the first one and I noticed that two of the shapes have 3 points and the other two have 4 points. Also two of the shapes have 3 sides and the other two have 4 sides. I noticed that the each of the two shapes that look similar are in common in many different ways.

Good ideas, Taylor! Ayan from the British International School of Houston in the USA said:

To match Venn 2 & Venn 3 I simply looked at the pairs. For example, on Venn 3 it shows two squares in one circle and on Carrol 2 it show the same two squares in the same box. This indicates that those two should be paired up.
Similarly, the same situation applies to Venn 2. Once Venn 2 and Venn 3 are paired up with their Carroll diagrams, there is only one Carroll diagram and only 1 Venn diagram left. Through process of elimination, we see that these two will have to match up.

Well done for working out which Venn and Carroll diagrams look similar, Ayan. We've uploaded a PDF of Ayan's solution.

I wonder if there are any different ways of working this out? If you have any ideas, please email us as we'd love to hear from you.