Collatz Sequence & Binary Analysis

Project Overview

This project explores algorithm implementation at the hardware architecture level using MIPS Assembly. It calculates the famous Collatz Sequence (also known as the $3n+1$ problem) for any given integer.

Beyond generating the sequence, the program performs a deep statistical analysis: for every number generated, it inspects its raw binary representation to calculate its Hamming Weight (the count of set bits / 1s), identifying which number in the sequence is computationally “densest”.

Multi-Language Support 🇮🇹 🇬🇧

The logic and documentation are available in both languages:

• English: Project_eng.txt
• Italian: Project_ita.txt

The Algorithm

The core loop implements the Collatz rules using efficient bitwise logic instead of expensive division/multiplication instructions where possible:

1. Input Handling: Accepts integers, automatically converting negative inputs to positive absolute values.
2. Sequence Logic:
   • If Even: Divide by 2 (Implemented via logical right shift srl or arithmetic logic).
   • If Odd: Multiply by 3 and add 1.
   • Termination: The loop halts when the sequence reaches 1.

Bitwise Analysis (Hamming Weight)

The unique feature of this implementation is the Binary Analysis module. For each number $X$ in the sequence, the program runs an inner loop that:

1. Iterates through all 32 bits of the register.
2. Uses masking (andi) and shifting (srl) to count how many bits are set to 1.
3. Compares the count with the current maximum ($s3) and updates the record holder ($s2) if necessary.

Example Output

Enter a number: 6
Sequence: 6 3 10 5 16 8 4 2 1 

Number of elements in the sequence: 9

Number with the maximum amount of ones in its binary representation: 5
It has a number of ones equal to: 2 (Binary: 101)

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MIPS Assembly Sum Calculator

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Low-level MIPS Assembly program that calculates min/max sum sets with manual memory management and arithmetic overflow protection.

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