How to Solve Edge Matching Cards Puzzles - Practical Advice

While I was at high school, around 25 years ago, I was first introduced to an edge-matching puzzle.  This is made up of 16 square card pieces, with a colour on each edge, that have to be arranged into a 4x4 square so that all the colours of the edges on each piece matches their adjacent neighbour.

It all sounds so easy.

I recently acquired two versions of this edge-matching puzzle - one based on cogs, and one based on footballers - with a view to finally solving it.  I didn't solve the puzzle while I was at school, and I've not revisited it much since, except to conclude that it was impossible, or was going to take longer to solve than I wanted to invest in it.  Well, things have changed, and as it's clearly been pestering me for 20+ years, so I figure it's time to rise to the challenge - especially since the Special Educational Needs Co-ordinator at my high school took delight in telling me that one of the 'weaker' children in our year group had successfully solved it.  Challenge accepted, Mrs Kirkham!

Here, for reference, are the 16 cards in the "Mechanical Mayhem Puzzle":

My first few attempts at the puzzle were not at all successful.  I very quickly connected a few pieces, but never made significant progress, and never really got past about 75% complete  before the pieces refused to go together any more.  I therefore decided it was time to get some hints and help, from that reliable source of all knowledge - the internet.

The most helpful information I found was from a research paper into edge-matching puzzles - apparently, these puzzles are not trivial, in fact they are N-difficult.  The advice was to find matching pairs, and then try to connect pairs to form fours, sixes and eights.  The pairs didn't always form squares, but at least I was now working systematically.  Here are a few candidate pairs, as examples:

And then some candidate squares:

As I became more familiar with the cards, I quickly identified patterns that would not work - for example there are only two squares which contain the bronze-coloured cog shown on the right.  Therefore they must form a pair, since the likelihood of them both being on the outside edge of the 4x4 square seemed low to me.  I didn't count each kind of cog and work out which were the most and least frequent, but I soon noticed that the large golden cog was very common, as was the yellow daisy-style cog.  The very small silvery cog was quite rare, so I was more careful with how I placed any cards that contained it.

And, with a little trial and error, this approach worked for me (with some adjustments, intuition and observation).  You can see from my final solution that the key pair of bronze coloured cogs are in the top-left corner, and yes, I conquered the edge-matching puzzle (and it only took me 20+ years).