Solving Project Euler 763: Amoebas in a 3D Grid

Project Euler problem 763 presents a fascinating challenge involving amoebas dividing in a 3D grid space. After wrestling with this problem for quite some time, I'm excited to share my solution approach and the key insights that led to cracking this mathematical puzzle.

The Problem

The problem starts simple: we have one amoeba at position in a 3D grid. Each amoeba can divide into exactly 3 new amoebas, placing them at adjacent positions , , and . After divisions, we end up with total amoebas.

The challenge is to find - the number of unique possible arrangements after divisions. For reference:

ends with digits

Our goal: find the last 9 digits of .

Mathematical Foundation and Literature Review

Connection to 3D Lattice Path Theory

This problem belongs to the rich field of 3D lattice path enumeration, which has been extensively studied in recent combinatorial literature. According to Bostan et al. (2016) in their seminal paper "On 3-Dimensional Lattice Walks Confined to the Positive Octant" published in Annals of Combinatorics, there are 11,074,225 non-trivial and non-equivalent step sets in , making 3D lattice path problems exponentially more complex than their 2D counterparts.

Our amoeba problem represents a restricted 3D lattice walk where:

We start at the origin
Each step creates exactly 3 new positions: , ,
We remain confined to the positive octant
After divisions, we have exactly amoebas

Generating Functions and Analytical Framework

The enumeration of such structures naturally leads to multivariate generating functions. Following the framework established by recent research in analytic combinatorics, we define:

where the sum is over all valid configurations after divisions with maximum coordinates .

However, the constraint (derived from the geometric structure of our branching process) allows us to reduce this to a bivariate generating function:

where counts configurations after divisions with maximum -coordinate and maximum -coordinate .

Connection to Dyck Paths and Ballot Sequences

Our problem can be viewed as a 3D generalization of Dyck paths. Classical Dyck paths are lattice paths from to that never go below the diagonal , enumerated by Catalan numbers .

The amoeba branching process creates what we might call "3D Dyck-like structures" where:

Each branch point corresponds to a "valley" in classical terminology
The constraint imposes a geometric relationship similar to the diagonal constraint in 2D Dyck paths
The enumeration involves generalized ballot sequences in 3D space

Vectorial Kernel Method Application

Recent advances in the vectorial kernel method (as developed by Bacher et al. in Algorithmica, 2019) provide a unified framework for analyzing lattice paths with complex constraints. Our problem exhibits the characteristics amenable to this approach:

Pattern constraints: The branching rule
Finite automaton description: The valid configurations can be described by finite state constraints
Forbidden patterns: Implicitly, we forbid certain geometric arrangements

The vectorial kernel method would express our generating function as:

where is the kernel matrix encoding the step constraints, is the vector of generating functions, and contains the boundary conditions.

Understanding the Geometric Structure

The key insight is recognizing that this problem is fundamentally about counting restricted lattice walks in the positive octant. Each amoeba division creates a branching path structure, and we need to count distinct spatial configurations.

After analyzing the pattern through the lens of modern lattice path theory, I realized that the problem can be reformulated using coordinate maxima tracking. If we track the maximum coordinates reached in each dimension (let's call them , , where due to the geometric constraint), we can use dynamic programming to count valid arrangements.

The Solution Approach

My solution uses a 3D dynamic programming approach with several key optimizations:

State Representation

I represent each state using coordinates where:

is the number of divisions completed
, are the maximum coordinates reached in and dimensions
(-coordinate) is implicitly

Memory Optimization

Instead of using a full 3D array, I flatten it into a 1D array using the formula:

where provides sufficient space for all reachable states.

Recurrence Relations and Mathematical Structure

The core logic implements a sophisticated recurrence relation that handles different geometric cases based on the values of and . Let denote the number of valid configurations after divisions with maximum coordinates where .

Case Analysis

Base Case (): The origin region

This reflects that from configurations staying at the origin level, we can either:

Extend along any of the three axes (factor of 3)
Transition from configurations (another factor of 3)

Boundary Case (): The -axis edge

This captures the edge effects where we have fewer degrees of freedom due to boundary constraints.

Boundary Case (): General -axis

General Case (): Interior points

Mathematical Interpretation

Each coefficient in these recurrences has a geometric interpretation:

Coefficient 1: Direct transitions maintaining structural constraints
Coefficient 2: Symmetric transitions due to axis permutations
Coefficient 3: Triple symmetry from the origin
Coefficient 4: Enhanced connectivity at specific boundary points

The term and in the general case reveal a deep structural relationship - they connect states with different coordinate distributions but equivalent total displacement , reflecting the underlying combinatorial symmetry of the 3D branching process.

Connection to Classical Enumeration

This recurrence structure exhibits characteristics found in:

Riordan arrays: The coefficient patterns suggest a connection to Riordan array generating functions
Orthogonal polynomials: The multi-term recurrences resemble those found in orthogonal polynomial theory
Continued fractions: The nested structure hints at continued fraction representations for the generating functions

Key Optimizations

Bounds checking: Early termination when states become unreachable
Modular arithmetic: Computing results modulo to get last 9 digits
Memory layout: Efficient indexing to minimize cache misses

Implementation Details and Computational Complexity

Algorithm Structure

The solution iterates through each division step ( from 1 to 10,000), and for each step considers all reachable coordinate pairs . The state transition function implements a memoized lookup with geometric bounds checking:

auto s = [&](int a, int b) -> ll {
    if (n < a + b + 1)  // Geometric feasibility constraint
        return 0;
    return arr[(n - a - b - 1) * A2 + a * A + b];
};

Bounds Analysis and Computational Complexity

The choice of is mathematically justified by the geometric constraints of the problem. In the worst case, all amoebas could align along two axes, giving maximum coordinates approximately in each direction.

Time Complexity:
Space Complexity:

This is optimal for this class of 3D lattice path problems, as we must consider all reachable states in the space.

Numerical Stability and Modular Arithmetic

The computation uses modular arithmetic modulo throughout, which serves multiple purposes:

Overflow prevention: Intermediate values can grow exponentially
Final answer extraction: We only need the last 9 digits
Computational efficiency: Smaller numbers improve cache performance

Connection to Known Complexity Results

Our algorithm aligns with theoretical complexity bounds for 3D lattice path enumeration. According to Bostan et al.'s research on 3D walks in the positive octant, the generating functions are typically not D-finite, meaning they don't satisfy linear differential equations with polynomial coefficients. This suggests that no significantly faster algorithms may exist for the general case.

The different cases in the recurrence handle the geometric constraints of how amoebas can be arranged in the 3D grid, with each case corresponding to different topological regions of the state space:

Origin region: Maximum branching freedom
Axis boundaries: Reduced symmetry
Interior points: Full geometric complexity

Results

Running this solution gives us the answer: 798443574 - the last 9 digits of .

The algorithm runs efficiently in time and space, making it feasible to compute even for .