I love doing puzzles and have considered how I could make my own for a while. I found that I could make laser cut puzzles if they were on the smaller side. If the puzzles are too big, the gaps caused by the laser become too big. This page shows you two different puzzles I've made with slightly different methods.
I found this picture online that was simple enough to use for a test puzzle design. I used this image to make the laser cut file in Adobe Illustrator.
This is the laser cut file. The laser cutter rastered the image in grayscale onto the board of MDF I used. I kept the puzzle assembled after I cut it and used acrylic paint to color the design that the machine outlined.
This is an image I used for a second puzzle. Since this is an actual photo, I did not want to paint it. I printed the photo onto sticker paper which I stuck to a sheet of corkboard.
I put the corkboard sheet into the laser cutter and had the machine cut out the design, cutting through both the photo layer and the corkboard layer. This is what the final puzzle looked like.