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'Jenny's Logic' printed from http://nrich.maths.org/
Jenny must know for sure that Tom's cards are either both even or both odd. The only way she can be sure of this is if she's selected the three even cards leaving Tom only odd numbered cards. Therefore Jenny has cards $2$, $4$ and $6$ and the sum of her cards is $12$.
This problem is taken from the UKMT Mathematical Challenges.