Mazement

Sheila was so pleased with her new trio of admirers and their puzzle that they were permitted to sit around her at the high table that night. Brian was slightly put out because Al got to sit between himself and Sheila; there are some disadvantages to being a trio rather than a duo.
Still, the jealous stares from all the other young (and old) bucks helped smooth his ruffled feathers a bit.
Between dinner and dessert Al showed himself to be a saint by offering to trade with Brian. "You did come up with the puzzle, after all," he murmured sotto voce.
Brian smiled slightly. "I thought you'd forgotten. All's fair in love and war, after all. Thank you, regardless." Sheila gave Al an especially brilliant smile at his self-sacrifice in trading places with Brian, making him turn strawberry and Bert growl.

The merriment went well into the evening. Sheila was not only beautiful but also had a wry sense of humor and cheerfully off kilter way of viewing thing. She kept her guests of honor in stitches with her comments and stories.
At length, though, the party broke up. The Alberts and Brian wound up sleeping in a hay loft because the innkeeper denied them lodging. He claimed to be full up, but they recognized him from the feast as one of Sheila's admirers, and put their rejection down to jealousy. They settled down philosophically enough, and slept soundly with visions of red-headed sirens dancing in their heads. The dance of the seven veils, no less.

Brian woke before the others, and slipped out without waking them. He was more than a little smitten with Sheila, and wanted to give her something nice that might help fix her regard. He had spied a shop with curiosities on the way into town, and he made his way there after snagging a bit of porridge at the inn. The cook was the only one up, and didn't realize that the innkeeper had blacklisted him.

He entered the curio shop to find himself menaced by teetering stacks of boxes and books and gizmos galore. Most of them he couldn't begin to identify.
A little tortoise of a man hunched behind the grimy counter, round head protruding from an overlarge collar on a thin neck. Brian half expected him to pull his head into his shirt at the sight of an actual customer.
Brian nodded politely, and started poking around with interest.
Most of the stuff was junk, but he did find one beautiful and intriguing object. It appeared to be a block of inlaid wood. It was solid except for a small round hole in one side, and as he tilted it back and forth he could hear something going clack . . . clack inside. Each time he tilted it there would be a pause and then the soft knocking sound of something hard thunking into the wood.

The shopkeeper was lured by this odd sound from behind the desk.
"Ah! Beautiful, isn't it?" he said with a sort of melancholy pride.
"It is! What is it, though?"
"Why, it's a maze with a little copper ball in it. The goal is to get the ball out.
"I made it myself, and used to be able to get the ball out without any mistakes, and then return it to its proper resting place in the maze. "But some kids came in and played with it, and I don't know where in the maze the ball is anymore. I can't solve it any longer. I don't think anyone can, except by the sheerest luck."
Brian studied it with interest. "What kind of maze? Any cycles? is it all in one dimension?"
"Oh yes, it's on a single level, just like you would draw it out on paper. The tunnels are just big enough for the ball to run down and turn corners comfortably. No cycles, though. I didn't think that was sporting."
"Hmm. Interesting. I'll take it, I think."
The little shopkeeper blinked. "But you won't be able to do it!"
"Oh, I think I could. Since there aren't any cycles I bet I can do it with the right series of moves. It helps that I can hear it knocking about in there."
The little shopkeeper stared at him. "But how?"
"I'm not quite sure; but I know you can get out of cycle-less mazes pretty easily under normal circumstances. This is harder of course since I can't see my options, and the ball might roll where I don't intend it to, but I think it would still be do-able. Besides, it isn't for me, it's for Sheila."
The little shopkeeper looked wistful. "You think she'd like it? But what if she can't do it?"
"She can, I'm quite sure."
With a cheerful wave Brian went out with his prize.

Who is right - Brian or the shopkeeper? Could someone consistently solve the hidden maze? If so, how?

Solution