Jigsaw Puzzles

Aging VW Bug overloaded with luggage

Apparently this is an AI generated image. The cartoon quality appealed to me. Never much cared for VW Bugs. Several of my high school friends had them, and I've ridden in them on occasion, but I've never had much use for them. Why spend $400 on a bug when I can buy a perfectly cromulant piece of Detroit Iron for $200? And back then highway cruising was essential, and doing that was a struggle for a bug.

I've been thinking about jigsaw puzzles. One item is how the size of the puzzle, as in the number of pieces, affects the odds of a match between two pieces, edge pieces specifically. Typically I pick out the edge pieces first. I place them on the line where I imagine the corresponding edges will lie. With a 300 piece puzzle I typically leave space on either side of the puzzle for four columns of pieces and a space for one or two rows above and below the puzzle. These side areas are not enough to contain all of the non-edge pieces so I'll end up with rows of extra pieces filling the interior of the puzzle. 

When I have completed the edges, the whole screen is occupied with pieces so it doesn't leave a lot of room to work, so we have to shuffle pieces until we have connected enough pieces that we have some room to work. It would be nice to have a bigger screen if the program didn't adjust the size of the pieces based on the screen size. Catch-22 there.

I start by going through the pile of pieces and picking out the edge pieces. I just place these edge pieces randomly on their corresponding line. When space gets tight on that line I will stop my sorting and see if any of the pieces on this edge will fit together. That's one nice thing about digital puzzles, it leaves no doubt as to whether the piece is correct or not. Get close enough to the position and it will snatch it right from your hand. It would probably snap/click too, but I have almost certainly turned the volume off. It will fool you though. Put it in a place where you think it goes and it won't complain. The sound would come in handy here. And then when you get to the end and the one piece in your hand won't go in the one spot left on the puzzle, you got fooled. Set the piece aside and pick up the whole puzzle and move it around. The incorrect placed piece will sit there and a new hole will appear in the puzzle.

Trying to keep on track here. Take a 300 piece puzzle, they come in two versions: 15 x 20 and 12 x 25. These have corresponding aspect ratios, i.e. the pieces are all roughly square in that if you measure the width of the puzzle and divide by the number of pieces in a row or column, the result will give the average width or height of a piece. Do that in both directions and you should get similar numbers. 

How to determine the width of a picture on a computer? Well, there is a digital width, but I get tired of poking around in obscure programs for niggling details. Do it the same way you would with a real cardboard puzzle - with a tape measure. Compare aspect ratios if you wish.

Take a 15 x 20 puzzle. You will be able to place at least ten pieces along the bottom edge without them crowding each other. At some point though the line will get crowded, maybe on the eleventh piece, maybe  not till the fifteenth, but it will eventually reach that point, so now I start looking for pieces that can combined and so eliminate one of the spaces and make some room. So, if you have ten pieces on a line that will hold 20, what are the odds that at least two pieces will match? How about the odds of all ten pieces finding a match, they might join up in pairs or triplets, but everyone finds at least one match. And what are the odds of all ten pieces joining into one long group?

The other issue is how the puzzle program works internally. Presumably they start with an image and then cut it up into pieces and send those pieces down the wire to the display controller with instructions on where to display them. Are they using a bit map to mark out individual pieces, or are they using vectors to draw the dividing lines? I imagine using a bitmap would be easier, but it would have to be scaled for the display. So you send the full size bitmap along with the image to the display controller, and the display controller is going to have to scale the image and the bitmap. Whether this is done in the CPU or the GPU would depend on their relative capabilities. Using vectors would just be computer nerds showing off.

Update the next morning. Corrected some typos, added a line or two.