The Math Behind Sudoku, in Plain Words

It starts with Latin squares

A Latin square is a grid where every symbol appears once in each row and once in each column. Sudoku is a Latin square with one extra rule, that each of the nine boxes also holds every symbol once. That single added constraint is what turns a simple grid into a puzzle worth solving.

How many grids exist

The number of complete nine by nine Sudoku grids was counted in 2005. It is 6,670,903,752,021,072,936,960, which is about 6.67 followed by twenty-one digits. If you ignore rotations, reflections and symbol swaps that produce essentially the same puzzle, the count of truly distinct grids drops to a mere 5,472,730,538. Either way, you will never run out of boards.

The magic number seventeen

For years people wondered how few clues a proper puzzle could have. In 2012 a team using a large computer search proved that the answer is seventeen. There exist puzzles with seventeen clues that have a single solution, and none with sixteen. It is a lovely result, part clever mathematics and part sheer computing power.

Why it is a hard problem in general

Ordinary nine by nine Sudoku is small enough for a computer to solve instantly. But the general version, played on an n by n grid, belongs to a class of problems that mathematicians call NP-complete. In plain terms, as the grid grows there is no known shortcut that stays fast. The nine by nine board sits in a sweet spot, hard enough to be interesting for a person, small enough to always be solvable.

Shapes do not change the math

Shapedoku swaps the digits one to nine for nine glowing shapes, but every count and every rule above still holds. The symbols are just labels. The logic underneath is exactly the same beautiful structure.