The p of ps - grid puzzle

Question

Please fill in all boxes with letters. (And what is the p of ps?)

Answer 1

The 'p of ps' is:

EUROPE

And the image should be completed like this:

Allow me to talk you through my thought process step-by-step...

Step 1:

The first thing of note here was that there are...

...exactly 26 boxes in the graphic in the top-right of the image. Since we are required to replace all symbols and spaces with letters it's not a big leap to suppose that we are intended to fill each of these 26 boxes with a different letter of the English alphabet.

How to do this though? My next thought was that the clusters of boxes might represent...

...words spelled with these letters, with the number of times that letter appeared being proportional to the size of its box outline.

Since the cluster second in from the left only had two unknowns I focussed on that word first, trying to work out what word might be spelled from the letter jumble {IILT##@}. I soon hit upon ITALIAN as a possibility. If we then substituted #=A and @=N in all other parts of the image, I then realised that the top letter jumble {AAACDINNNS.} had a related answer: SCANDINAVIAN! It seemed I was on to something...

Step 2:

I now started to think about the theme - what was it exactly? Clearly it would be something related to...

...geography, culture or languages, and I turned my attention to the bottom-right {ACIMN$*} jumble. For a long while I got sucked into the language route, considering this to be either ROMANIC or ARMENIC...

Then another realisation struck me. What if the central cluster was supposed to indicate...

...EUROPE? This would fit with whatever theme I was currently uncovering. For this to be the case, we would have to have *=E (ruling out ROMANIC) and $=R. ARMENIC it was, then... (At least for now...)

Step 3:

At this point I realised the error of my ways...

Looking at the bottom-left jumble {AEIINR~} and the one two to its right {AALN^~} I spotted that if ~=B and ^=K we could spell IBERIAN and BALKAN - these weren't language-related at all; they were European peninsulas! This meant my 'ARMENIC' was actually CRIMEAN, and I had inadvertently pinpointed E and R whilst having completely misidentified what they represented!

Step 4:

Now armed with the knowledge of the theme, I was able to resolve the final jumble:

{ADLNT?.} was JUTLAND, with ?=U since this also had to appear as a letter in 'EUROPE'.

But how to disambiguate the final two? This was where the top-right grid came in again...

Noting that so far all of the letters were appearing within their '0', '1', '2', '3' and '4' categories in alphabetical order from left to right, I reasoned that the arrow beneath it must be suggesting this ordering. But without knowing what these numbered categories actually are, we're in a bit of a pickle...

This was when I turned my attention fully to working out what this grid meant. And then an idea struck me...

I surmised that this had to be related to some physical property of the letters. And after staring at the screen a while I noticed that there was another connection to the theme here - the letters had been organised into categories based on the number of 'peninsulas' in the letter, with a 'peninsula' here being a 'loose end' on the outside of the letter. For example, the fully enclosed 'B' had no loose ends at all, 'Q' had just one at the bottom, and 'K' had a whopping four of them!

_{Note from OP, added later: Technically, this puzzle involves counting something like "loose line segments" rather than "loose ends", as hinted by the key and the (unexpected) inclusion of the letter I in the '1' category of the grid, as remarked later in this answer:}



By this logic, it was possible to see that +=O and %=P, since an 'O' has no 'peninsulas' and a 'P' has one, putting them in categories '0' and '1', respectively.

Step 5:

Finally, time to fill the remaining spaces in the grid.

Knowing the property governing the category definitions and that letters were ordered alphabetically, this was now a straightforward task:



Note that this precise categorisation relies to some extent on the typeface used - for instance, the absence of a top bar on a 'J' (giving just two rather than three loose ends). The inclusion of 'I' in the '1' category is perhaps controversial - to my eye it has two loose ends, not just one, but never mind... The puzzle is now solved, and we have found our 'p of ps', or 'peninsula of peninsulas' - a nickname for the continent of Europe!

Bonus:

It's also worth pointing out that if you look at...

...a map of Europe, the clustered arrangements also somewhat resemble the shapes of these peninsulas and their relative positions on the map. Their arrangements have not been chosen at random!