Farmer Barton shooed Lee away from Marilyn, affection in his eyes. He matched words to the action, just to pretend that he wasn't alone.
"Now, Lee, I've told you before. You can't eat the chicken. I'll need the eggs on this trip, and I need her alive to barter with when we get to Woden. Go hunt yourself down some field-mice."
The fox cocked its head, appraising Barton for a moment, then obediently moved off the side of the trail, shadowing the farmer and his hen as they walked. An hour later, when Barton stopped at the bank of the river, Lee returned, her mouth spotted with blood. Barton had put down his heavy bag of feed and was frowning at the water.
There was no sign of the bridge. He wondered if he'd come to the wrong place, but no, he'd followed the trail. The recent disasters, it seemed, had torn the bridge down, and the river had erased all evidence that it had ever been.
He couldn't go back. There was nothing left of his home. And to leave the trail, to strike out into unknown regions in this chaotic time, was to court death. The frontier was everywhere now.
Then he spotted it! A tiny canoe, lodged in the bank. Some traveler, more skilled than himself, must have carved it, after the bridge was gone but before Barton arrived. Barton hurried over to inspect it, but turned back as he heard an ominous tearing. Marilyn had gotten into her feed. She'd torn open the bag, and was gobbling it down as though she thought she'd never get another chance. Barton yelled--that feed had to last her many more days yet!--but it was Lee who chased the chicken off, nearly catching her. Barton scooped Marilyn up before the fox could get at her, or she at the feed.
The farmer sighed, hoisted the feed-bag, tear-side up so it wouldn't spill, and started again toward the canoe, then froze. The cold logic of the situation had arrived. The canoe was only big enough for him and one other--the fox, the hen, or the feed.
He'd have to take three trips, with two returns, but no matter what order he did it in, he'd have to leave either the fox and the hen, or the hen and the feed, alone together at some point. If he left the fox and the hen behind, most likely that would be the end of Marilyn before Barton could return. If he took the fox first, the hen would be left behind with the feed. She'd eat too much of it, maybe all of it, which would also mean her death soon enough on this long journey. And Marilyn couldn't die. Lee was a beloved pet, but Marilyn was his last valuable possession; without her, Barton himself was most likely lost.
There was no choice. The cold equations of mass and volume, the harshness of the place and time, dictated everything. Barton seized the fox, relying on her tameness, her trust in him, and dashed her head against a rock on the bank. He buried her shallowly, with a swift economy. Then he hoisted the feed into the canoe, and began the first of two trips across the river. He hid his tears. Not because there was anyone watching; just so he could pretend that he wasn't alone.
The unsolvable problem is a powerful tool in fiction. At minimum, you get pathos from it. You can use it to set up a wrenching choice, wherein our insight into the character is deepened by what she decides to sacrifice, or the character evolves from the trial.
Awkwardly, the reader is typically a human, a strange creature who, when faced with an unsolvable problem, often solves it. This tends to undermine the pathos, distract from the actual plot, and--worst of all--means that the reader has beaten the author. Authors are also typically humans, and they hate losing. They'll often end up solving the problem themselves, which can lead to an incongruously happy ending that sabotages the emotional impact. This too is annoying to the reader.
In realistic fiction, the rules are generally well-defined enough that a reader will accept that a problem that appears to be insoluble truly is. But in speculative fiction, where the rules are always poorly defined compared to realistic, this awkwardness shows up more often than it doesn't. Any time an episode of Doctor Who has an unhappy ending, online fora will instantly fill with fans indignantly explaining that this tragedy could have been avoided; and the hero, a super-smart alien, should have seen the loophole. You can't rule out every single loophole in dialogue, because in speculative fiction anything is possible, and with enough viewers, anything will seem plausible to someone. The only way I know of that a speculative fiction author can create a true no-win scenario is to declare that The Very Laws of the Universe Demand That This Must Be, and You Can't Fight Fate.
As a reader, what I hate is when I can see a solution that clearly didn't occur to the author, and that still involves pathos, drama, character development, et. cetera. Of course, I need to complain about this at length whenever it happens, because it means that I've beaten the author, at least in my own mind. Ask me some time how His Dark Materials should have ended (everyone has a different idea, but mine is the best one). But in this space I'll restrain myself to one general pattern that needs to go away. I've seen at least two stories, The Matrix Reloaded and the short story Endosymbiont, that use it blatantly. Several more that do similar things, sometimes with hand-waving excuses. It's this: a person who is also a computer program is faced with two doors. Go through one, and continue living as before: a true person, though a digital one. Go through the other, and become the contemporary kind of computer program: useful, but no longer truly a person. They're being asked to heroically sacrifice their independent existence for the good of the world. The fatal flaw with this idea is that programs can be copied. Go through both doors. It's still a sacrifice, because half of you experiences dying, or unlife, or enslavement, or something. But it's better than going through just one.