Sudoku
Constraint propagationHTML UISearch
Sudoku are the 9x9 number puzzles where you starting from a certain number of
filled in cells the player tries to fill all remaining cells with the numbers
from 1-9 under the constraint that no number appear more than once in any
given row, column, or any of the nine 3x3 sub-boxes of the 9x9 grid.
If you want, you can build a UI to let people play sudoku in which case you might want to provide an easy way to set up the board and maybe some ammenities like indications of what numbers are still available in each square.
Another maybe more interesting project is to write a sudoku solver, that is a program that takes a sudoku puzzle in some form and figures out the solution. There are a number of ways to tackle this problem. A brute force search that tries every possibility in a systematic way will, in theory, ultimately solve any sudoku. And for easy puzzles may solve them quite quickly. But harder puzzles can take quite a long time to solve with brute force so you may need to be more clever. Peter Norvig, the Head of Research at Google has written an excellent essay about solving Sudoku. The code is in Python but the ideas can be implemented in Javascript. He also provides several very hard puzzles to test out the efficiency of your solver.
Note, if you want, you can write a solver without any UI that you will run from the command line. I will need to teach you a few things about how to run Javascript outside of the browser and accessing data from disk but none of it is hard.