Overview

Pips is a puzzle game developed by the New York Times that challenges players to cover a grid-like shape with a provided set of dominoes, adhering to specific constraints related to the sum of pip counts in each section of the puzzle. Players can solve puzzles across three difficulty levels—easy, medium, and hard—released daily. The game implementation leverages F#, a functional programming language well-suited for handling complex algorithms without side effects.

Game Mechanics

The objective of Pips is to strategically place dominoes such that the total number of pips in designated regions matches required totals. Each puzzle setup is unique and may have varying solution complexities ranging from a single solution to thousands.

Problem Solving Approach

This project employs a backtracking algorithm, which is an effective method for solving combinatorial problems. The algorithm systematically explores all possible placements of dominoes:

1. Place a domino on the board.
2. Verify that the current configuration meets all game constraints.
3. When encountering an invalid placement, backtrack and attempt a different arrangement.

Additional optimizations to enhance solving speed include:

• Geometric Tiling Optimization: Utilizing geometric properties to limit potential placements of dominoes. For example, understanding that it is impossible to fill an odd-numbered space with dominoes can reduce search complexity significantly.
• Pruning Techniques: Conditions are checked throughout the placement process, allowing the algorithm to abandon infeasible paths early, thus optimizing performance.

Solution Validation

Regions within the puzzle are categorized into types with unique constraints:

• Equal Regions: All cells must contain the same pip count.
• Unequal Regions: All cells must contain distinct pip counts.
• Sum Constraints: Regions that require the sum of cell values to be less than, greater than, or exactly equal to a target value must also be validated.

Results and Performance

As of now, the solver has successfully addressed all 88 hard puzzles published between August 18 and November 13, 2025, with an average solution time of about 1.8 seconds. The most complex puzzle encountered, featuring an elephant shape, was solved in just over a second, showcasing the efficiency of the implemented algorithm. Furthermore, some puzzles yielded numerous solutions, with one reaching an astounding count of 2,764,800.

Code Structure

The repository includes detailed implementations for essential components:

• Domino: Represents the basic unit of gameplay.
• Cell & Edge: Structures for managing board coordinates and adjacent cell relationships.
• Board: Maintains the entire state of the puzzle, including placed dominoes and cell values.
• Region: Enforces specific rules governing how dominoes can be placed in various parts of the grid.
• Backtrack Module: Implements the algorithm that traverses the puzzle configuration space to find solutions.

To view the implementation and contribute to the project, visit the Pips GitHub Repository for complete source code and additional documentation.