"Oi, Hermione."
"We heard you admired our Headless Hats as sort-of clever."
"So how would you improve them?"
Hermione looked up from reading her book on her favorite squashy couch in the Common Room. Twin-freckled grins hung over her. She blinked!
Ron spoke up, "She may be the brightest witch in our class, but she's still muggle-born and only beginning to learn magic. You two are two years ahead of us!"
But Arthur's sons were not easily deterred.
"Well, we thought a fresh approach..."
"A different point of view, you know..."
"Might add something interesting!"
Harry spoke up, "The only one who could really USE a Headless Hat is Sir Nick, and I don't know any way to give him one. None of the rest of us really want to join the Headless Hunt. It's just a joke item, fun but pointless."
"That's a fair point!"
"A ghost is neither here nor there..."
"Neither dead nor alive..."
"And how would a hat be alive, let alone die?"
But Ron had noticed Hermione's eyes glazing over, and tugged on the nearest twin's arm, pointing at her. They ceased their banter, suddenly listening, with a serious expression.
"There's Schr dinger's Cat." Crookshanks jumped up on her lap and purred, thinking she wanted him. The four boys were baffled. What cat? But Hermione was doing what she always did, rummaging through her bookbag. In a moment she held up - a book! "My parents sent me this as a bit of light reading." The displayed title was : In Search of Schr dinger's Cat: Quantum Physics and Reality by John Gribbin. (A/N This is a real book.) She looked at the four puzzled boys and started to simplify. "The point is that no one knows what is in a sealed box until you open it. You might rattle it, or weigh it, but you don't know for sure. Now, if you put a cat in a box with a container of poison that has exactly a 50% chance of killing the cat (Crookshanks hissed and stalked off), you don't know until you open it what has really happened to the cat. Totally hypothetically, the cat could be immortal, or not. So, if you sealed a really nice hat in a box with a very breakable container of acid, or something, that could totally destroy the hat, wouldn't that be a Schr dinger's Hat - or in other words, a Ghost Headless Hat, hypothetically neither here nor there? Would that do it?"
The Twins were looking at her in wonder. Ron was totally lost, and Harry (who had a slight idea of what Quantum Physics was) was almost as amazed as the Twins.
"Could we borrow this book, oh brilliant one?"
"Yes." She handed it over, "Mind, I DO want it back! It was a Birthday present!"

The next night after dinner the Twins were waiting for Harry, Ron, and Hermione. One Twin was carrying a large wrapped gift box. Once in the Common Room, they gathered more Griffs around the central table. "Now, we need Sir Nick, and I didn't see him at dinner. Anyone know how to summon him?"
"You called?" Sir Nick drifted through the mantle over the fire, hovering near the table.
"We have a gift for you, a token of our esteem!"
"You will need to open the box to show it to me," said the puzzled Ghost, "as I cannot really touch it."
"No, THIS gift cannot be opened!"
"We will need to put the box away, somewhere safe where no one will ever open it."
"But YOU should be able to take out the gift, because if we did it right it's a Ghost Hat."
"Specifically, it's one of our best Headless Hats, your proper style and feathers, and very strongly charmed."
"Please, see if you can pull it out?"
Everyone in the room was silent, watching. Sir Nick looked at them all, and then reached through the side of the box. His hands went into the box briefly, and then he pulled them back together - holding a beautiful insubstantial hat! He gasped slightly. It had worked! With tredipation he turned the hat and set it upon his head - which vanished!

The next dark-of-the-moon night the Griffs were all at the windows of their tower, with the lights off behind them. Across the sky above the Forbidden Forest soared the Headless Hunt. Normally a scene to frighten the timid, tonight the Hunt seemed brighter than the stars, and the wildest Rider whooping across the sky was waving a beautiful new Hat.