Phrase for a problem that resolves itself shortly after giving up

I feel like this is a not uncommon series of events when I’m faced with a problem:

• I have a problem.
• I try to fix the problem, to no avail.
• I resign myself to the problem being unsolvable.
• Shortly afterwards, the problem is resolved with minimal effort on my part.

For example, suppose I’ve lost a set of keys. I spend a while searching the flat for them, to no avail. I give up, sit down and perhaps call somebody to whinge about it – and then spot them sitting in front of me. Complaining about it is often the precursor to discovering it’s been fixed.

Is there a phrase that describes this feeling, of a problem that solves itself shortly after you’ve given up?

I’ve often said Murphy’s law or sod’s law of the universe to describe this sort of event, but I don’t think that’s accurate – this isn’t a bad thing (far from it!), it’s just a shorthand to say “it’s a weird quirk of life”.

I considered the term heisenbug, but (1) that seems a bit technical, and (2) the problem is solved, not reproducing intermittently. To me, a heisenbug is still a concern even if it doesn’t reproduce on demand, because it might recur later. That’s not the case here.

Perhaps this is some form of irony? I was thinking perhaps situational irony, but I’m aware that this is a heavily overused term, and I’m not sure if it’s accurate here.

I’ve tried googling, and looking back through old phrase-request questions, to no avail. I’m not sure if there’s a term I’m not finding, or if this is just less common than I think.

The scenario you describe is attributable to Humphrey's Law, which states that 'a normally automatic or unconscious activity is disrupted by consciousness of it or reflection on it'.

Of course, this assumes that the reason you don't find your keys is that you're too consciously focused on the act and outcome of finding your keys to allow the instinctive and unconscious mechanisms of your brain to do what they need to do to actually find them - and that the reason they eventually turn up is that this impediment of consciousness is removed.

But we all know the real reason: the universe is laughing at you.

Which, of course, is Sod's Law.

I think with a concept like this you're going to get different phrases depending on the specifics of the situation.

For example, in the case of the keys, a common phrase would be that it was staring me in the face

to be obvious The answer to this problem was staring him in the face, although at first he couldn't see it.

thefreedictionary.com

However, this phrase only really applies if the specific solution to the problem was something obvious that you simply missed because you were looking to hard.

Let's say we took a different approach. Say you were working on a chemistry experiment for ages and you simply couldn't get the solution you were looking for. You give up, forget to wash out your equipment overnight, and come in the next day and see that your desired result was achieved, simply by leaving it overnight. You gave up on the problem and it fixed itself, but the solution wasn't blindingly obvious. In that case, 'staring me in the face' would not apply.

If you're looking for a descriptive phrase, consider the following in the context of your first example regarding problem-solving: "incubating the problem".

You’re walking down the street, completely relaxed, and you are not thinking about any particular thing. Then all of a sudden the solution to a problem you’ve been working on for weeks pops into your head out of the blue. You wonder why you didn’t think of it before. You’ve experienced your subconscious mind at work. Your subconscious mind will continue to work on a problem long after you leave it. This is known as incubating the problem. Many idea people report that their best ideas come when they are not thinking about their problem. - Michael Michalko, How to Get your Subconscious Mind Working on a Problem (emphasis, mine)

If you're looking for an idiom, consider:

Stop searching and you will see. - attributed to Lao Tzu, Getselfhelp.co.uk

These kinds of sayings are open to many interpretations. The one I'm trying to highlight is that of finding a long-lost, forgotten object when simply sweeping under the cupboards one day.

One might possibly refer to your situation as a "tardy serendipity" or a "delayed serendipitous solution". Wikipedia notes that the word "serendipity" '...was coined by Horace Walpole in 1754. In a letter he wrote to a friend, Walpole explained an unexpected discovery he had made by reference to a Persian fairy tale, The Three Princes of Serendip. The princes, he told his correspondent, were "always making discoveries, by accidents and sagacity, of things which they were not in quest of".' The resolution of the problem is "serendipitous" because the quester has abandoned the search for a solution when one presents itself spontaneously. The adjective "tardy" recognizes the effort of the previous search and the fact that the seeker could have been spared some labor had the solution presented itself earlier rather than "tardily".

The OP's situation might be described in colloquial English (BE?) as "jammy".

Urban Dictionary: Jammy - ...lucky, defying probability with outrageous good fortune.

So, one might say: "Well lo and behold. There they (the keys) were all the time. Sitting on the table, right in front of my nose. How jammy is that?"