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Collatz Conjecture

Tue, 15 Jul 2025 18:03:03 +0900

Is it true that “any number eventually becomes 1”? ── Playing with the Collatz Conjecture

Hello! I’m kenji.

Suddenly, but if you hear “a rule where any number eventually becomes 1”, isn’t it a bit mysterious?

For example, 19, or 87, or even 1000000. If you tweak the numbers according to appropriate rules, for some reason it converges to “1” at the end.

Such a dream-like story is the Collatz Conjecture.

What is the Collatz Conjecture anyway?

First, let me introduce the rules.

Start: Choose any positive integer
Operation:
- If it is even → Halve it (n → n / 2)
- If it is odd → Triple it and add 1 (n → 3n + 1)

If you repeat this forever, the conjecture says that any number will eventually reach 1.

For example, starting from 6:

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6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It properly became “1”. Welcome back!

Let’s do it in code: Collatz in Python

Now, in times like this, it’s faster to try it in code! Let’s output the “Collatz sequence” in Python.

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def collatz(n):
 steps = [n]
 while n != 1:
 if n % 2 == 0:
 n = n // 2
 else:
 n = 3 * n + 1
 steps.append(n)
 return steps

# Example: Starting from 19
print(collatz(19))

When you execute it:

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[19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

It splendidly reaches 1. Even though it takes quite a detour, it firmly reaches the goal at the end!

By the way, even if you start from 29, it reaches 1 in the same way.

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print(collatz(29))

When you execute it:

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[27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242,
121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350,
175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167,
502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479,
1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911,
2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732,
866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35,
106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1]

Surprisingly, it takes 111 steps!

Moreover, there are scenes where it balloons to over 9000 along the way. It’s a pattern that takes a huge detour before reaching the goal.

So, what’s amazing about it in the end?

What’s amazing about this conjecture is,

Even though it hasn’t been proven, it seems to become 1 no matter what number you use

That’s the point.

Eh? Then, what about 1 trillion, or 10 quadrillion…?

If you thought that, you are sharp. Actually, it has been verified up to about “2 to the 68th power” using computers, and all have reached 1. Unbelievable…

But, it hasn’t been theoretically proven that “it always happens”. This is what they call an “unsolved problem” in the world of mathematics.

Who is Mr. Collatz?

So, reading this far, you might wonder “who is Collatz anyway?”. Let me introduce him properly!

Name: Lothar Collatz
Nationality: Germany
Year of birth: 1910 - 1990
Title: Mathematician (Active in the fields of functional analysis and number theory)

He proposed this conjecture in 1937, and since then, for over 80 years, no one has been able to prove or disprove it.

By the way, this problem is so simple yet so deep that even Paul Erdős (a super famous mathematician) is said to have said this:

“Mathematics may not be ready for such problems.”

In other words, the theory that human mathematics hasn’t caught up with this mystery yet…

No “complex math formulas” are necessary

The good thing about the Collatz Conjecture is that anyone can play with it.

You can do it if you have paper and pen. If you write code in Python, you can test it automatically. And yet, cutting-edge mathematicians are seriously challenging it.

Doesn’t it make you excited?

Bonus: Code to test it all at once

I’ll also include code to test various numbers all at once.

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for n in range(1, 21):
 steps = collatz(n)
 print(f"{n}: {steps} (Steps: {len(steps)-1})")

This outputs the Collatz sequences from “1 to 20” all at once.

Conclusion: This world is indeed mysterious

So, that’s the Collatz Conjecture.

Even though it’s super simple
No one can prove it
It’s a huge problem in the math community

It’s an existence like a cluster of mysteries.

Even programming beginners can try it, so please definitely play with it~!

Advantages and Disadvantages of Creating Programs with Win32API + C++

Sat, 12 Jul 2025 12:30:35 +0900

The Appeal and Challenges of Developing with Win32API + C++

For those who want to master Windows application development, Win32API + C++ remains a powerful option today. This combination, which allows for interacting with the OS at the closest distance, combines high speed and flexibility.

On the other hand, since it is vastly different from modern development styles, learning it requires dedication.

On this page, from the perspective of an active Windows app developer, we explain its advantages and disadvantages in an easy-to-understand manner.

Advantages

Ultra-Fast Native Execution

Because C++ and Win32API operate at the layer closest to the OS, there is almost no unnecessary overhead. It boasts overwhelming execution speed and highly efficient use of CPU and memory.

High Flexibility and Freedom

You can minutely control all aspects of an application yourself, such as window control, asynchronous processing, COM integration, and process management. It’s also possible to build purpose-built tools and your own custom frameworks.

Easy to Distribute Without Runtimes

Since external runtimes like .NET or Java are unnecessary, it can be distributed as a single executable file. Troubles during redistribution are less likely to occur, and it’s appealing that it can easily run without an installer.

Capable of Creating Lightweight Apps

Because it requires only a bare-minimum configuration, a major feature is its extremely small memory footprint. It runs comfortably even on low-spec PCs or in virtual machine environments.

Advanced OS-Level Control is Possible

Global hooks for mouse and keyboard, fine-tuning of window styles, manipulation of system menus, and other forms of control that are difficult with standard languages and libraries can be realized.

Disadvantages

Low Development Efficiency

GUI construction must also be done entirely in code, and creating a single button can sometimes require dozens of lines of code. Modifications during design changes are also cumbersome, making productivity lower compared to developing with UI frameworks.

Maintainability Tends to Decrease

The code often has a specialized structure, such as message loops and window procedures, leading to issues with readability and reusability. It also has aspects that make it unsuitable for team development and long-term maintenance.

Difficult to Support Modern UI

Supporting UX expected in recent years, such as high DPI support, touch interfaces, accessibility, and dark mode, is difficult. Each must be handled manually one by one, which requires a lot of effort.

No Cross-Platform Support

Because it is a completely Windows-exclusive API, it cannot be ported to macOS or Linux. If you are planning multi-platform deployment, you will need to select other technologies.

Extremely High Learning Cost

You must understand concepts and mechanisms that are rarely used today, such as handles, GDI, COM, and OLE. Much of the documentation is old, requiring time and perseverance to learn.

Suitable Uses

Lightweight tools such as file launchers and hotkey assistants
System utilities like clipboard manipulation and IME control
Native control applications like global hooks and window captures
Driver auxiliary tools that work closely with hardware

Unsuitable Uses

General consumer apps where modern UI/UX is important
Prototyping and MVP development built with a priority on speed
Large-scale projects intended for long-term operation and team development
Cross-platform products that need to support multiple OSs

Evaluation Summary

Aspect	Evaluation
Execution Speed	◎ Very Fast
Memory Efficiency	◎ Excellent
Development Speed	× Slow
Maintainability	× Low
Cross-Platform Support	× Not Supported
Modern UI Support	× Weak
Freedom of OS Control	◎ Overwhelmingly High

Conclusion

Win32API + C++ is a tool suited for developers who want to “handle everything in the OS themselves.” While its power is immense, learning and operating it requires a corresponding level of dedication.

Whether it’s worth “daring to choose” it depends entirely on the nature of the app you are aiming for.

Diving into the world of #include <windows.h> without relying on GUI frameworks or modern languages―― That choice still remains meaningful today.

Collection of Fonts Useful for Programming

Sat, 30 Mar 2024 02:21:31 +0900

JetBrains Mono

An open-source programming font provided by JetBrains.

Download: https://www.jetbrains.com/ja-jp/lp/mono/

HackGen

HackGen is a programming font that combines the English programming font Hack with Genjyuu Gothic, a derivative of Source Han Sans.

Download: https://github.com/yuru7/HackGen/releases

Ricty Diminished

Ricty Diminished is a sister font to Ricty. Instead of Inconsolata and Migu 1M, it is synthesized using Inconsolata and Circle M+ 1m with the Ricty generation script. Since it does not include IPA Gothic glyphs, the number of available kanji glyphs is limited, but in exchange, it can be distributed under the SIL Open Font License.

Download: https://rictyfonts.github.io/diminished

Cica

It uses Hack + DejaVu Sans Mono for alphanumeric characters and symbols. For other characters, it uses Rounded Mgen+. It is a programming font designed for coding.

Download: https://github.com/miiton/Cica/releases

References:

Migu

A synthesized font of M+ and IPA.

Download: https://itouhiro.github.io/mixfont-mplus-ipa/migu/

Osaka

A font that allows you to use Osaka, a standard Mac OS font, on Windows.

Download: http://ifs.nog.cc/osakattf.hp.infoseek.co.jp/

PlemolJP

A programming font that combines IBM’s open-source Japanese font IBM Plex Sans JP with IBM Plex Mono, also from IBM’s Plex series.

Download: https://github.com/yuru7/PlemolJP/releases

[By Language] Summary of Online Code Execution Environments

Sun, 09 Apr 2023 14:01:00 +0900

Python

Google Colaboratory

Google Colaboratory is an online execution environment for Python.

GPU usage is possible.
With Colab notebooks, you can combine executable code and rich text (images, HTML, LaTeX, etc.) in a single document.

It is based on Jupyter Notebook.

Rust

Rust Playground

JavaScript

Javascript Playground

Java

paiza.IO Online Java compiler

Haskell

Haskell Playground

Swift

Online Swift Playground

C++

C++ 14

Others

AWS Cloud9

An online integrated development environment provided by Amazon AWS. Tools for over 40 programming languages such as Node.js, JavaScript, Python, PHP, Ruby, Go, and C++ are pre-packaged.

Visual Studio Code for the Web

An online development environment provided by Microsoft. However, languages that require compilation cannot be executed or debugged.

paiza.IO

An online execution environment supporting 24 major languages such as C, C++, Java, Ruby, Python, PHP, Perl, etc.

CodePen

An online execution environment for HTML, CSS, and JavaScript.

codeanywhere

An online IDE. Allows code sharing and chatting.