Every time someone asks me "what are the shiny brown things?" that they got from the ground underneath the Ohio buckeye (Aesculus glabra), I wonder what they did as kids in the fall. I grew up in England where we had horse chestnuts (Aesculus hippocastanum), very similar trees but bigger and with spiny fruit casings instead of the smoother ones in the picture. The fruit inside look the same though.
We would collect hundreds of the fruit, stomping on the spiny shells to get to the fruit inside. This was not the same as our sweet chestnut (Castanea sativa) collecting trips where we would collect the fruit for roasting; the fruit of the horse chestnut are not edible. Instead we would drill holes through the middle & hang them on shoe-laces; one fruit, or conker as we called them, per shoe-lace.
Then the fun began. At school or at home we would play conkers, a game in which you try to smash your opponent's conker with yours--taking it in turns to be either the one dangling your conker to be hit or the one swinging your conker as hard as possible to smash the other.
A conker with no victories would be a none-er, with one victory a one-er, two a two-er and so on. If you beat a conker with a number you got to add that number to yours.
It was very competitive, you would not want to risk a high number conker against a none-er, it was not worth it. Everyone had their secret recipe to improve the conkers, we soaked ours in vinegar to strengthen them--though they did not look so pretty or shiny after and it probably really made little difference.
We have several buckeyes in and around the gardens, the three most people notice are between South African Plaza and Sacred Earth. That said, my favorite one for good conkers is not on the grounds and I am not sharing where it is! If I can beat the squirrels to the fruit I still collect a few just in case I meet someone who wants to play.