Been there
My kids have both gone through major bragging phases.
It's interesting because one of the things that I've learned as I've gotten older is how much I DON'T know (I used to be really arrogant, lol). I can spend time with a person whose IQ is 40+ points lower than mine and still learn something from their perspective.
This is what I've tried to teach my kids: It doesn't matter how "smart" you are - everyone has something to contribute. Also, anyone can
say they're smart, but it's acting smart by making good choices that really counts.
When they were too young to understand that, I'd simply say "bragging is rude - don't do it." When they'd ask why, I'd say "it makes other people feel bad and they won't want to be friends with you." My son seemed to accept this more readily than my daughter, who would get so caught up in the fact that her age peer didn't understand something, etc., that she didn't really hear me.
Another thing I tried (this seemed to work better with my daughter) is to say that "different kids learn things at different times." For example, "you're further ahead in math than (insert name), but she's better at swimming than you are." THIS she accepted.
I think it depends on how you view it as a parent. For instance, who gives a hoot about my IQ - the fact is, I'm terrified of heights, and of airplanes. Pilots and Flight Attendants A.M.A.Z.E me. I don't for the life of me know how they do it. How can I brag about some arbitrary cognitive test score when I turn into a blubbering mess at 100 feet off the ground? Good grief. Smarter is NOT always better. There's more to life than smart.
"Yes you could play chess on your 6th birthday, but (insert name) has a beautiful voice... can we please stop talking about Chess for a minute so we can enjoy her singing?"
That's the key I think - teach them that everyone has some kind of strength. Then it no longer comes across as bragging, but rather "sharing strengths."
