Ha-ha-ha, apparently I'm now a world-building tool :-)

But I think the world is established enough. At least the way I read the first sentence is that the noose is there as a regular part of the playground equipment, not something the girls have brought there. Thus, if in their world playgrounds have hanging nooses as a regular part of playground equipment, it is established that kids killing themselves for fun is a regular occurrence in their world, and it is not happening in our world. Then there's casual nudity too.

Anyway, I love the story and would love to see more too :-)

I would, however, strongly advise against writing it directly into the message box, if you don't want to lose your story to something as silly as captcha, or eventually to gurochan dying again in another couple of years as it always does. Write in MS Word or something, then, first of all it will not go anywhere as long as you save regularly. Second, it would also feel more substantial and incentivize you to write more substantial stories, as you wouldn't have to write them in one go and would have more time to think about them and flesh them out better. For example, in this story the first part looks all good, you've probably thought it out at least somewhat before you started writing it. The second part, however, feels more rushed and could have benefitted if you let your thoughts first brew properly in your head, which would work better if you write them down into a doc file (or your preferred text processor).

This story also reminds me of a few things.
First is a story about Rei and Asuka from Evangelion having a somewhat similar bet, only they were to hang each other with their strength, not themselves. Was this story inspired by that?
Then, I had another idea I never used, where the rope would be properly tied to something, but the noose would be set up low enough for the girl to easily be able to reach the floor with her feet, so in order to hang she has to bend her legs behind her back and then hold onto her ankles with her feet (see the picture).
Though, while we're at it, I imagine some sort of a yoga pose with your feet behind your head could be used too.
Feel free to use or not use these ideas however you like.