To see jigsaws grouped by piece count and theme, from first puzzles for toddlers to harder pieces for adults, take a look at the jigsaw collection at jigsaw puzzles.
If you like the Genius Square, the same puzzle idea runs through a small family of games. Genius Star adds a bonus mechanic and a different set of pieces, Genius Gems and Genius Connection change the shapes and the goal, and Bee Genius is pitched at younger children who are new to shape puzzles. Each one keeps the roll-and-fill pattern while giving you a fresh challenge.
If you want a shortlist of games tested for family play and grouped by how long they take, browse the family and party games chosen by brain teaser puzzles.
The 3D and giant multi-storey versions bring something new to the classic. Instead of a flat board, counters travel up and around a tall structure, with the snakes and ladders running between levels, which children find far more exciting to watch. Larger garden versions scale the whole thing up for outdoor play, so the game moves from the kitchen table to the lawn on a sunny day.
For a shortlist of compact games that pack down small and set up fast, browse the travel-friendly games chosen by puzzle games for adults.
For gift ideas sorted by age band,
recent Wikipedia blog post with the number of players and rough playing time listed for each, browse the games-by-age pages at puzzle games for adults.