Should I challenge a Virgo man and show him that I am not easy to catch?

Can you play hard-to-get with a Virgo man?

I don't think Virgo's all that interested in a challenge, unless he's trying to improve his skill-set. He may analyze the most efficient way to chase a not-easy-to-catch woman, but he's more into improving his game than winning the prize.

In other words, I think your plan will backfire. You've got two things working against you: his element is Earth and his mode is Mutable. Earth signs are not exactly noted for their immediate, passionate responses -- they are more cautious and slow to act. And Mutable signs disperse energy -- your Virgo man has difficulty mobilizing himself to pursue you. Mutable signs are mentally-focused -- he may just think about how he'd catch you.