Let me clarify okay? I got something out of your post that I don't think other got. I may have read it wrong though.
Your friend had your daughter put on a hat to go to an activity.
Lets say it was a green hat to make things easier.
The friends child put on her siblings soccer hat to wear.
They both had on hats at this time.
Some time during the activity they were playing and tumbling around and the friend wanted to wear the green hat and your child to wear the soccer hat.
Let's say during their play time both hats fell off. When it was time to go your child went and picked up the green hat she had been given to wear to the game.
The friends child did not go get the soccer hat they had been given to wear...correct????
The friend is mad because your child did not remember to remind her child to pick up the hat her own child was supposed to have on?
In that case I would remind my friend that her daughter had taken the soccer hat to wear and that your child had been given the green hat to wear and they had returned the green hat they had taken at the beginning.
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So I guess I am with you, how is it your daughters fault that your friends daughter didn't bring back the hat "she" took to the activity.
Why is it even anything to do with your daughter? If her daughter took the hat then ended up forgetting it why is this lady even talking to you about it being your daughters problem.
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I guess if the friendship is worth more than the cost of replacing the hat, if it's even possible at this point, then I guess I would offer to pay for half and if that wasn't enough to appease her I would say that you did way beyond your part and let the friendship be a thing of the past.
I would also go to the field and see if the hat made it to the lost and found or if it is still laying somewhere under the seats or on the ground somewhere. Looking for it would be okay.
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Just to mention something it's not too early to teach your child.....
this is how head lice get spread from child to child. Sharing hats and dress up clothes is pretty yucky when you think about it.