No … Errr, and yes.

I've tested water and alcohol-based markers, both consumer and professional brands, Sharpies, colored pencils, and crayons and nothing "sticks" to the pieces due to the lamination protecting the puzzle pieces. Generally speaking, everything rubs off on your fingers with the slightest moisture and will create a huge multi-colored mess on everything you touch including your clothing. I've not yet tested acrylic paint.

The only type of marker that will leave a mark on the puzzle pieces are "wet erase" markers like EXPO vis-à-vis markers or Staedtler Lumograph markers which are designed to apply to nonporous surfaces like laminates and glass. These markers are brilliant for coloring my puzzles and after you're done you can use a lightly dampened cloth to wipe away your work, box the puzzle for storage, or start coloring it all over again …

However, there are two caveats you need to consider (there is always a caveat):

1. While wet markers leave a mark that is much more durable than dry erase markers, they can still be smudged by a fingertip even after they dry—ANY moisture at all will mottle, smear, or remove the color.
    
   
2. (Carefully) Cleaning or erasing the puzzle: You will be able to easily wipe away your coloring work with a lightly dampened cloth (Note: if you can wring water out of your cloth, it's far too wet—You're wiping off laminated paper, after all.) 
   
   You will NOT be able to remove all of the color—Color will remain lodged in micro ridges and imperfections in the lamination. In other words, you will never be able to return the puzzle to pristine white if you decide to color it with wet erase markers.
   
   DO NOT use alcohol to remove residual color as it will dissolve both the lamination and black puzzle ink.
   
   
   

If you decide to use wet erase markers to color the image, as long as you are very careful, you could frame the puzzle behind glass to protect your masterpiece, but this comes with a final caveat ... Wet erase marker ink may fade over time, especially if exposed to heat or sunlight.

Hopefully this information helps:) If you decide to color your puzzle, please post your creation to Instagram and tag @OddGoods or send me a photo directly!

Cheers,
Paul