Can anyone figure this out?
Yes... took me about 3 minutes. If that.
Start with the 2x2=4 part and it all pretty much follows from there. 2 must represent a single digit other then itself which when multiplied by itself isn't itself and yet the answer is a single digit. This rules out 0, 1, 2, 4, 5, 6, 7, 8, 9 so you know 2 represents 3. That then tells you 4 must represent 9. And if 4x4=67 then you also know what 6 and 7 represent. It all follows about as easily as that.