When the party has died down, Isabella, for one, is well and truly exhausted. She explores the palace until she finds a room with a bed in it, and into this bed she flops, still in her clothes and holding her staff and carrying the cordial in her pocket. She sleeps late, because the party kept her up so late and she hadn't really slept the night before; but around noon, she stirs, and gets up, and goes looking for James and wherever her backpack may have got to. The backpack she finds in the great hall where the principal mass of the party was; some enterprising creature took both bags from the battlefield at Beruna up to the castle for them, and she only wishes she knew who it was. She takes her bag to her room and carries James's with her and continues looking for her friend.
"I might literally write down 'kissing the king'. But nobody looks at my notebook without me right there supervising the page-turning so we're probably safe."
"Also it would not be on a schedule page so much as on a thinking-about-things page. Pleasant things."
"See, I do most of my equivalent of that playing with my puzzle," says James. "Which I can't really write things down in, at least not in recognizable words."
"I'm not at all clear how you go about thinking with the puzzle," says Isabella.
"Yeah. I know. It's not really very explainable. Sometimes it's just something to do while I think about other stuff, but sometimes I use it to help with the math."
"My thinking isn't very mathy," remarks Isabella, sitting on the foot of her bed.
"A lot of mine is. And even the stuff that isn't, I can express a lot of it as math if I try."
"What does it mean to express something that isn't math as math?"
"Well... I'm not exactly sure. I mean, I know how I do it, but any way I try to explain it just seems to amount to 'I play with my puzzle and think about things'."
There is considerably more of that in the days that follow, sprinkled in with all their usual activities, behind closed doors; they do wind up telling Viridian the wardrobe-nymph, as well as the two senior housekeepers who'd be likeliest to stumble on them and Mr. Tumnus who is likeliest to need to explain their whereabouts, for convenience, but otherwise remain quite discreet.
He has tried throwing himself on a bonfire (his meltwater put it out) and beheading himself (it grew back) and putting a sword through his heart (it fell out, after an unknown amount of time spent frozen, dreaming vague frosted dreams). He has tried drinking molten lead (the worst part was cutting it out of his stomach after it cooled) and impaling himself on a fixed vertical spike, in case gravity helped keep it in place (evidently it didn't help enough), and crushing himself under the biggest rock he could arrange (he spent a long, long time being very, very uncomfortable until his successive icy healings produced a pile of crushed ice uneven enough for the boulder to roll off). Around the turn of the year, he tries building an oven and shutting himself inside, reasoning that at least this way his meltwater won't put out the fire. It drowned him instead, which didn't work either. After about a day, he gave up and climbed out.
He still gets thirsty. Perhaps he could die of that.
So he finds himself a remote little cave, as dry as possible, and curls up inside and resolutely does not go out to find snow to eat or water to drink.
...Winter does recognize this rat. Winter wishes he didn't recognize this rat. Winter wishes the rat had found a different damn cave.
He doesn't really want to—there's nothing left for him among Jadis's creatures, without Jadis—but no, there is one thing. Jadis's creatures will speak to him. Jadis's creatures will not flee him in terror or turn him over to the Aslan-blessed human rulers of Narnia.
"Lead me there," he says at last.