I actually suspect that my DD is the way she is partly because she has realized that she is very smart and that she often notices or remembers things that others (including her parents) do not. I think she doesn't trust us to get it right, to some extent.
This is part of why we always explain our rules to DD7... because in the course of explaining them we're going to show her how we're aware of things that she's not. It tells her the value of listening to us, because we're pretty smart. Deep down, kids want to know that their parents are sharp and are going to make good decisions, because it gives them a great sense of security. If they're convinced their parents don't know what they're doing, that's a scary feeling.
We also make sure we don't always lose to her on purpose... partly to teach her that we're good at stuff, too, and thus worthy of her respect, but also to teach her how to lose with grace. We wouldn't want to turn her loose on the school playground without having experienced defeat before.
Last week she wanted to play "Guess Which Hand?" with a little rubber ball, and I guessed correctly each time, which totally amazed her. Then it was my turn to hold the ball, and she guessed wrong every time. She was still having fun, but she simply could not believe how she couldn't get it right at least once through sheer luck. So I explained it to her:
"In both cases, I knew something you didn't."
"What?"
"When you had the ball, your hands were different, because in the one with the ball your knuckles bulged out a bit."
[she experiments, observing both hands next to each other as one holds the ball, tries holding it different ways, bulging out her other hand to match, etc.]
"What about when you had the ball?"
"When I did it, the ball wasn't in EITHER hand."
[quick demonstration of how I'd drop the ball in my lap, show her the one hand she'd requested, and pick up the ball with the other]
She thought that was HILARIOUS. But then, trying to put one past each other is a regular part of our daily family life.
So yeah, don't be afraid to show off. Little kids want to believe that their parents are pretty amazing.