I tend to just ask the mom if their child is sick. I don't think it's rude at all. It doesn't bother me if a mom asks me if my child is sick. Like you said, they could just have allergies or young kids could be teething. It'd be a shame to miss out on an event because you think someone is sick. If the child IS sick, maybe that mom would get the idea that you shouldn't take a sick child out into public!

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