Darel Rex Finley in 888

Collatz Conjecture Analysis (But No Proof; Sorry)

2012.01.06   prev     next

ALAS, I find I am unable to develop a proof of the Collatz Conjecture. But in my attempts to do so, I have come up with a few interesting ways of analyzing the problem, that perhaps are worth sharing.


A “Hailstone” sequence goes like this: Start with any positive integer, say, 52. If the number is even, divide it by 2, otherwise multiply it by 3 and add 1. Since 52 is even, we will divide it by 2, and get 26. Now, do the same thing again. Since 26 is even, we will divide it by 2 and get 13. Now, 13 is odd, so we will multiply it by 3 and add 1, resulting in 40.

Starting from 52, the hailstone sequence does this:

52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, etc.

Notice that the sequence settled into a three-number cycle: 1, 4, 2. This repeats forever.

Collatz Conjecture

The Collatz Conjecture says that no matter what positive integer you start with, your hailstone sequence will, in a finite number of steps, wind up repeating at 1, 4, 2. The challenge is to find a rigorous logical-mathematical proof that the conjecture is true.


Let’s lay out the positive integers into a quadrant grid like so:

  19  38  76  152  304  608  1216
  17  34  68  136  272  544  1088
  15  30  60  120  240  480   960
  13  26  52  104  208  416   832
  11  22  44   88  176  352   704
   9  18  36   72  144  288   576
   7  14  28   56  112  224   448
   5  10  20   40   80  160   320
   3   6  12   24   48   96   192
   1   2   4    8   16   32    64

The left column is all the odd numbers (1, 3, 5, 7, etc.), and each row is a doubling sequence. Except for the left-most column, all the numbers are even, so at first glance, it appears that this arrangement contains mostly even numbers. But appearances are deceiving: The chart contains every positive integer, and contains each one exactly once. This is true because:

  • All the odd numbers are in the left column, with no repeats.

  • Since any even number can be converted to an odd number by dividing it by 2 one or more times, and all the odd numbers are included, then all the even numbers must be included.

  • Since any one doubling sequence (row) does not include the same value twice, and each row starts with a unique odd number, no even number can be included more than once.

Therefore the layout contains each counting number exactly once.

Because each row is a doubling sequence, the hailstone step of dividing an even number by 2 is equivalent to moving one column to the left. Therefore, if you are currently on any even number (i.e. on any column to the right of the left-most column), you simply move straight to the left until you reach the left-most column.

Since the left-most column is composed of odd numbers, when you are on it, you must multiply by 3 and add 1. This takes you to an even number in another row — unless you are on the number 1 (the bottom row), in which case it takes you to the number 4 in the same row. We know that the bottom row is the only row whose odd number goes to the same row because each row consists of numbers of the following form:

  n  2n  4n  8n  16n  .  .

This sequence does not include 3n, whose value falls in-between 2n and 4n. 3n+1, however, can reach 4n — but only if n is 1. So only the bottom row can go to itself. All other rows go to a different row than themselves.

Now we can think of the hailstone sequence as dealing with only odd numbers. Each odd number represents a row, and each odd number jumps to another row (i.e. to another odd number). The only odd number that jumps to itself is 1. (If we start the hailstone sequence on an even number, we can simply pretend that we started on the odd number that begins that number’s row.)

So our example sequence that started on 52 can be simplified to:

  13, 5, 1, 1, 1, etc.

(The conjecture is not yet proven, because we have not eliminated the possibility that two or more odd number are connected in an infinite loop, nor the possibility that a hailstone sequence could grow without limit.)

Every third row consists of numbers that are divisible by 3 (i.e. the rows that start with 3, 9, 15, 21, etc.). These rows can be discarded as effectively not part of the problem, because the hailstone sequence jumps to a new row with the operation 3n+1, which can never be divisible by 3. Hence, the rows 3, 9, 15, 21, etc. can serve only as a starting row for a hailstone sequence, but are never visited again. So we can simply pretend that a hailstone sequence that started on an odd multiple of 3 actually started on the next value in that sequence.

So the problem can be reduced to showing that all odd numbers that are not divisible by 3 eventually settle on the value 1. Those values are:

  1, 5, 7, 11, 13, 17, 19, 23, 25, etc.

(This looks like a set of prime numbers, but it isn’t — note the presence of the number 25.)

When an odd number n is subjected to 3n+1, it projects onto all other rows via a curve that looks like this:

In this diagram, hailstone moves from the 7’s row to the 11’s row via 22.  11 is approximately 1.5 times 7.

This curve hits only one row at a column value, and it cannot hit at the left-most (odd) column since 3n+1 is always even. Therefore, if it hits a higher row, it must hit that row’s second column (the row’s first even column). Simple math indicates that the new n will be approximately 1.5 times the old n. So when (odds-only) hailstone goes up from its current row, it goes up to approximately 1.5 times the value of the old row. When it goes down, however, it can go down any distance at all, even jumping straight to the bottom row. Note: If it could be proven that any (odds-only) hailstone sequence cannot indefinitely maintain an average percentage value change greater than the average for all odd numbers, then that might constitute a proof of the conjecture.


The formula for an odd number n that returns to itself in, say, four (row-jumping) steps is:

      abc + 3ab + 32a + 33
  n = ————————————————————
           abcd - 34

where a, b, c, and d are powers of 2 (not less than 2) that represent how much the number must be divided by to move it to the left-most column after landing in a new row.

The formula for an odd number n that returns to itself in five (row-jumping) steps is:

      abcd + 3abc + 32ab + 33a + 34
  n = —————————————————————————————
                abcde - 35

And so on.

If you set the values of all the power-of-2 variables (a, b, etc.) to 4, then the formula evaluates to 1, indicating that 1 is always a positive integer solution to the formula, and that it jumps to the value 4n (i.e. 4) with each row-jumping step.

For any of these formulas, if you could find any other combination of power-of-2 values that give an n value that is a positive integer other than 1, then (provided those power-of-2 values actually work with the hailstone rules), you would have disproven the conjecture by the discovery of an infinite loop that does not involve the bottom (n=1) row.

On the other hand, if you can prove that these formulas have no positive integer solution other than 1, then you have proven that there are no loops except at the bottom row.

It turns out to be fairly easy to show that if the power-of-2 variables are all at least 4, and any of them are greater than 4, then n evaluates to a number between 0 and 1. So the only cases we need to worry about are those in which one or more of the power-of-2 variables have the value 2. Playing around with these cases shows that n can be made negative (but we don’t care about that, only positive integers), and it can be made into positive values greater than 1, but they don’t appear to be integers. If there is some way to prove that they can’t be integers, then loops are eliminated except at the bottom row.

If There Are No Loops...

If you are able to prove that there are no loops except at 1, would it seal the conjecture? Maybe. You would still have to address the possibility that a hailstone sequence could become larger and larger without limit. There might be a way to show that that is impossible, that goes something like this:

Reverse hailstone is where you either multiply n by 2, or subtract 1 then divide by 3. The latter step, n′ = (n-1)/3, may be performed only if n is even and n-1 is divisible by 3, but the former step, n′ = 2n, may be performed on any value. So, reverse hailstone can be performed indefinitely (and it branches).

Reverse hailstone that starts on any row that is not the bottom row, cannot reach the bottom row, because if it could, then the bottom row would not be stuck in a forward-hailstone loop, and we know that it is.

If we already knew that a loop (other than at the bottom row) is not possible, then if forward hailstone starting at x grows without limit, then what does reverse hailstone starting from x do? To avoid a loop, it too must grow without limit — and not just by use of n′ = 2n — its row-jumping branches must grow without limit, otherwise the rows will become exhausted and a loop will be discovered. So there would have to be a hailstone sequence that comes down from infinity, then turns around and goes back up to infinity. My intuition tells me that this cannot be, but I have no formal proof.

Prime Factors

Another way to analyze Collatz is via the prime factorization of a number. Suppose you’re currently on a row that starts with n. n can be represented as a set of prime numbers multiplied together. We know that the numbers 2 and 3 won’t be among those primes, because n is odd and we’ve already eliminated divisible-by-3 rows from relevancy. So the prime factors will all be 5 or greater.

Now we move from n to another row, which starts with n′. Like n, n′ does not have 2 or 3 among its prime factors. Furthermore, it does not have any of the prime factors that belong to n. We know this because the +1 step gets rid of all of the prime factors of n. The steps to get from n to n′ are:

a. Multiply by 3: This step appends 3 to the prime factorization, but otherwise doesn’t change it.

b. Add 1: This step kills all the current prime factors (including the 3), because if a number is divisible by x, and you add 1, then it’s no longer divisible by x. One or more 2s appear in the prime factorization, because the number is now even. Other prime factors, not present before this step, may appear (or may not).

c. Divide by 2 until the number is odd: This simply removes all the 2s that appeared in the previous step.

So with each row-jumping step, the number changes from a prime factorization that included no 2s or 3s, to a new prime factorization that also doesn’t include 2s or 3s, nor does it include any prime factors from the old row.

True But Unproveable?

In reasearching the Collatz Conjecture, I have found some material that says that a more general set of problems has been shown to be unproveable; yet the specific (above-discussed) Collatz Conjecture may yet be proveable. I have also found some material suggesting that the Collatz Conjecture may one day be found to be true-but-unproveable.

Call me uneducated, but I find the whole concept of “true but unproveable” to be inherently contradictory. Yes, I can conceive that a conjecture such as Collatz’s might be true (i.e. there might be no exceptions to it) and yet there might be no logical/mathematical proof of that conjecture for us to find. However, if that is the case, then it seems only logical to state that we will never know it. If the Collatz (or any other) Conjecture is true but unproveable, then our knowledge of the Conjecture will always be that we don’t know for sure whether it’s true or not, and that we haven’t found any exceptions, nor any proof that there are no exceptions.

In other words, if you can show that conjecture C is true-but-unproveable, then you have shown that C is true, and therefore you have proven it. Which means it’s not unproveable.

Now, of course, we might one day discover a proof that the Collatz Conjecture can’t be proven. OK, I can imagine that. But in that case, we will never know whether there isn’t some exception to the conjecture lurking in the set of counting numbers. (Unless/until we find one.) This proof of unprovability won’t eliminate that possibility; it will just prove that we can’t prove there aren’t exceptions. Any claim that includes “the conjecture is true” is tantamount to a proof, and thus the conjecture loses unproveability.

Confused? Here’s some clarification. Suppose that the following propositions happen to be true:

  • The Collatz Conjecture is true; i.e. there are no exceptions to it. All positive integers behave as it posits.

  • There is no logical/mathematical proof that the Collatz Conjecture is true.

  • There is a logical/mathematical proof of the Collatz Conjecture’s unproveability.

In that case:

  • We will never find an exception to the Collatz Conjecture.

  • We will never know for sure that there isn’t an exception to the Collatz Conjecture.

  • We may possibly discover the proof that the Collatz Conjecture is unproveable.

  • We will never know that the Collatz Conjecture is “true but unproveable” (even though it is) — we will always have to wonder if it is unproveable because it is false, and we just haven’t found an exception yet.

Now, suppose that we define “true but unproveable” more precisely, to mean “true, but unproveable via technique set X?” OK, that might make more sense. But if a person claims to have shown that the Collatz Conjecture is:

(a) true, and
(b) unproveable via technique set X

then I’m only dimly interested in claim (b), while very interested in claim (a). Who cares about technique set X? If you’ve convinced yourself that there are no exceptions to the Collatz Conjecture, then I’d really like to know the specifics of how you convinced yourself of that. Whether the truth of the conjecture can also be established by some more limited set of techniques “X” is a far less interesting topic.

That’s All, Folks

And that’s all I’ve been able to figure out about the Collatz Conjecture. Don’t know if any of it can be developed further into a proof. I hope it was interesting.


Update 2012.01.18 — “Prime Factors” section added

Update 2012.08.20 — “True But Unprovable?” section added


Update 2014.10.21 — This analysis is continued here.


prev     next




Hear, hear

prev     next

Best Recent Articles

Method of Implementing A Secure Backdoor In Mobile Devices

When Starting A Game of Chicken With Apple, Expect To Lose

How I Clip My Cat’s Nails

Seasons By Temperature, Not Solstice

It’s Not A Criticism, It’s A Fact

Features (Regularly Updated)

A Memory of Gateway — news chronology of Apple’s ascendancy to the top of the technology mountain.

iPhone Party-Poopers Redux and Silly iPad Spoilsports — amusing litanies of industry pundits desperately hoping iPhone and iPad will go away and die.

Embittered Anti-Apple Belligerents — general anger at Apple’s gi-normous success.


My books

Now available on the iBookstore!



Daring Fireball

The Loop



Red Meat

Despair, Inc.

Real Solution #9 (Mambo Mania Mix) over stock nuke tests. (OK, somebody made them rip out the music — try this instead.)

Ernie & Bert In Casino

Great Explanation of Star Wars

Best commercials (IMO) from Superbowl 41, 43, 45, 46, and 47

Kirk & Spock get Closer

American football explained.

TV: Better Call Saul; Homeland; Survivor; The Jinx; Breaking Bad; Inside Amy Schumer

God’s kitchen

Celebrity Death Beeper — news you can use.

Making things for the web.

My vote for best commercial ever. (But this one’s a close second, and I love this one too.)

Recent commercials I admire: KFC, Audi

Best reggae song I’ve discovered in quite a while: Virgin Islands Nice

Pinball Arcade: Unbelievably accurate simulation of classic pinball machines from the late ’70s through the ’90s, with new ones added periodically. Like MAME for pinball — maybe better.

d120 dice: You too (like me) can be the ultimate dice nerd.

WiFi problems? I didn’t know just how bad my WiFi was until I got eero.

Favorite local pad thai: Pho Asian Noodle on Lane Ave. Yes, that place; blame Taco Bell for the amenities. Use the lime, chopsticks, and sriracha. Yummm.

Um, could there something wrong with me if I like this? Or this?

This entire site as a zip file — last updated 2018.02.01

Previous articles

Nothing More Angry Than A Cornered Anti-Apple

Let ’Em Glow

The Ultimate, Simple, Fair Tax

Compassion and Vision

When Starting A Game of Chicken With Apple, Expect To Lose

The Caveat

Superb Owl


Basic Reproduction Number

iBook Price-Fixing Lawsuit Redux — Apple Won

Delusion Made By Google

Religion Is A Wall

It’s Not A Criticism, It’s A Fact

Michigan Wolverines 2014 Football Season In Review

Sprinkler Shopping

Why There’s No MagSafe On the New Mac­Book

Sundar Pichai Says Devices Will Fade Away

The Question Every Ap­ple Naysayer Must An­swer

Apple’s Move To TSMC Is Fine For Apple, Bad For Samsung

Method of Implementing A Secure Backdoor In Mobile Devices

How I Clip My Cat’s Nails

Die Trying

Merger Hindsight

Human Life Decades

Fire and the Wheel — Not Good Examples of A Broken Patent System

Nobody Wants Public Transportation

Seasons By Temperature, Not Solstice

Ode To Coffee

Starting Over

FaceBook Messenger — Why I Don’t Use It

Happy Birthday, Anton Leeuwenhoek

Standard Deviation De­fined

Not Hypocrisy

Simple Guide To Pro­gress Bar Correctness

A Secure Backdoor Is Feasible

Don’t Blink

Predictive Value

Answering the Toughest Question About Disruption Theory

SSD TRIM Command In A Nutshell

The Enderle Grope

Aha! A New Way To Screw Apple

Champagne, By Any Other Maker

iOS Jailbreaking — A Perhaps-Biased Assessment

Embittered Anti-Apple Belligerents

Before 2001, After 2001

What A Difference Six Years Doesn’t Make

Stupefying New Year’s Stupidity

The Innovator’s Victory

The Cult of Free

Fitness — The Ultimate Transparency

Millions of Strange Dev­o­tees and Fanatics

Remember the iPod Killers?

Theory As Simulation

Four Analysts

What Was Christensen Thinking?

The Grass Is Always Greener — Viewing An­gle

Is Using Your Own Pat­ent Still Allowed?

The Upside-Down Tech Future

Motive of the Anti-Ap­ple Pundit

Cheating Like A Human

Disremembering Mi­cro­soft

Security-Through-Obscurity Redux — The Best of Both Worlds

iPhone 2013 Score Card

Dominant and Recessive Traits, Demystified

Yes, You Do Have To Be the Best

The United States of Texas

Vertical Disintegration

He’s No Jobs — Fire Him

A Players

McEnroe, Not Borg, Had Class

Conflict Fades Away

Four-Color Theorem A­nal­y­sis — Rules To Limit the Problem

The Unusual Mo­nop­o­list

Reasonable Projection

Five Times What They Paid For It

Bypassable Security Certificates Are Useless

I’d Give My Right Arm To Go To Mars

Free Advice About Apple’s iOS App Store Guidelines

Inciting Violence

One Platform

Understanding IDC’s Tablet Market Share Graph

I Vote Socialist Be­cause...

That Person

Product Naming — Google Is the Other Microsoft

Antecessor Hypotheticum

Apple Paves the Way For Apple

Why — A Poem

App Anger — the Supersized-Mastodon-In-the-Room That Marco Arment Doesn’t See

Apple’s Graphic Failure

Why Microsoft Copies Apple (and Google)

Coders Code, Bosses Boss

Droidfood For Thought

Investment Is Not A Sure Thing

Exercise is Two Thirds of Everything

Dan “Real Enderle” Ly­ons


Ignoring the iPod touch

Manual Intervention Should Never Make A Computer Faster

Predictions ’13


Zeroth — Why the Century Number Is One More Than the Year Number

Longer Than It Seems

Partners: Believe In Ap­ple

Gun Control: Best Ar­gu­ments

John C. Dvorak — Translation To English

Destructive Youth

Wiens’s Whine

Free Will — The Grand Equivocation

What Windows-vs.-Mac Actually Proved

A Tale of Two Logos

Microsoft’s Three Paths

Amazon Won’t Be A Big Winner In the DOJ’s Price-Fixing Suit

Infinite Sets, Infinite Authority

Strategy Analytics and Long Term Ac­count­a­bil­i­ty

The Third Stage of Computing

Why 1 Isn’t Prime, 2 Is Prime, and 2 Is the Only Even Prime

Readability BS

Lie Detection and Psy­chos



Microsoft’s Dim Pros­pects

Humanity — Just Barely

Hanke-Henry Calendar Won’t Be Adopted

Collatz Conjecture A­nal­y­sis (But No Proof; Sorry)

Rock-Solid iOS App Stability

Microsoft’s Uncreative Character

Microsoft’s Alternate Reality Bubble

Microsoft’s Three Ruts

Society’s Fascination With Mass Murder

PlaysForSure and Wikipedia — Revisionism At Its Finest


Patent Reform?

How Many Licks

Microsoft’s Incredible Run

Voting Socialist

Darwin Saves

The Size of Things In the Universe

The Self-Fulfilling Prophecy That Wasn’t


Nobody Was In Love With Windows

Apples To Apples — How Anti-Apple Pundits Shoot Themselves In the Foot

No Holds Barred

Betting Against Hu­man­i­ty

Apple’s Premium Features Are Free

Why So Many Computer Guys Hate Apple

3D TV With No Glasses and No Parallax/Focus Issues

Waves With Particle-Like Properties

Gridlock Is Just Fine

Sex Is A Fantasy

Major Player

Why the iPad Wannabes Will Definitely Flop

Predators and Parasites

Prison Is For Lotto Losers

The False Dichotomy

Wait and See — Windows-vs-Mac Will Repeat Itself

Dishonesty For the Greater Good

Barr Part 2

Enough Information

Zune Is For Apple Haters

Good Open, Bad Open

Beach Bodies — Who’s Really Shallow?

Upgrade? Maybe Not

Eliminating the Im­pos­si­ble

Selfish Desires

Farewell, Pirate Cachet

The Two Risk-Takers

Number of Companies — the Idiocy That Never Dies

Holding On To the Solution

Apple Religion

Long-Term Planning

What You Have To Give Up

The End of Elitism

Good and Evil


How Religion Distorts Science

Laziness and Creativity

Sideloading and the Supersized-Mastodon-In-the-Room That Snell Doesn’t See

Long-Term Self-De­lu­sion

App Store Success Won’t Translate To Books, Movies, and Shows

Silly iPad Spoilsports

I Disagree

Five Rational Coun­ter­ar­gu­ments

Majority Report

Simply Unjust

Zooman Science

Reaganomics — Like A Diet — Works

Free R&D?

Apple’s On the Right Track

Mountains of Evidence

What We Do

Hope Conquers All

Humans Are Special — Just Not That Special

Life = Survival of the Fittest

Excuse Me, We’re Going To Build On Your Property

No Trademark iWorries


Twisted Excuses

The Fall of Google

Real Painters

The Meaning of Kicking Ass

How To Really Stop Casual Movie Disc Ripping

The Solitary Path of the High-Talent Pro­gram­mer

Fixing, Not Preaching

Why Blackmail Is Still Illegal

Designers Cannot Do Anything Imaginable

Wise Dr. Drew

Rats In A Too-Small Cage

Coming To Reason

Everything Isn’t Moving To the Web

Pragmatics, Not Rights

Grey Zone

Methodologically Dogmatic

The Purpose of Lan­guage

The Punishment Defines the Crime

Two Many Cooks


One Last Splurge

Making Money

What Heaven and Hell Are Really About

America — The Last Suburb


What the Cloud Isn’t For

Diminishing Returns

What You’re Seeing

What My Life Needs To Be

Taking An Early Re­tire­ment

Office Buildings

A, B, C, D, Pointless Relativity

Stephen Meyer and Michael Medved — Where Is ID Going?

If You Didn’t Vote — Complain Away

iPhone Party-Poopers Redux

What Free Will Is Really About

Spectacularly Well

Pointless Wrappers

PTED — The P Is Silent

Out of Sync

Stupid Stickers

Security Through Nor­mal­cy

The Case For Corporate Bonuses

Movie Copyrights Are Forever

Permitted By Whom?

Quantum Cognition and Other Hogwash

The Problem With Message Theory

Bell’s Boring Inequality and the Insanity of the Gaps

Paying the Rent At the 6 Park Avenue A­part­ments

Primary + Reviewer — An Alternative IT Plan For Corporations

Yes Yes Yes


Hey Hey Whine Whine

Microsoft About Microsoft Visual Microsoft Studio Microsoft

Hidden Purple Tiger

Forest Fair Mall and the Second Lamborghini

Intelligent Design — The Straight Dope

Maxwell’s Demon — Three Real-World Ex­am­ples


Entitlement BS



Einstein’s Error — The Confusion of Laws With Their Effects

The Museum Is the Art

Polly Sooth the Air Rage

The Truth

The Darkness

Morality = STDs?

Fulfilling the Moral Du­ty To Disdain



Real Design

The Two Rules of Great Programming


The End of the Nerds

Poverty — Humanity’s Damage Control

Berners-Lee’s Rating System = Google

The Secret Anti-MP3 Trick In “Independent Women” and “You Sang To Me”

ID and the Large Had­ron Collider Scare

Not A Bluff

The Fall of Microsoft

Life Sucks When You’re Not Winning


The Old-Fashioned Way

The Old People Who Pop Into Existence

Theodicy — A Big Stack of Papers

The Designed, Cause-and-Effect Brain


IC Counterarguments

The Capitalist’s Imaginary Line

Education Isn’t Eve­ry­thing

I Don’t Know

Funny iPhone Party-Poopers

Avoiding Conflict At All Costs

Behavior and Free Will, Unconfused

“Reduced To” Ab­sur­dum

Suzie and Bubba Redneck — the Carriers of Intelligence

Everything You Need To Know About Haldane’s Dilemma

Darwin + Hitler = Ba­lo­ney


Designed For Combat

Speed Racer R Us

Bold — Uh-huh

Conscious of Con­scious­ness

Future Perfect

Where Real and Yahoo Went Wrong

The Purpose of Surface

Eradicating Religion Won’t Eradicate War

Documentation Overkill

A Tale of Two Movies

The Changing Face of Sam Adams

Dinesh D’Souza On ID

Why Quintic (and Higher) Polynomials Have No Algebraic Solution

Translation of Paul Graham’s Footnote To Plain English

What Happened To Moore’s Law?

Goldston On ID

The End of Martial Law

The Two Faces of Ev­o­lu­tion

A Fine Rec­om­men­da­tion

Free Will and Population Statistics

Dennett/D’Souza Debate — D’Souza

Dennett/D’Souza Debate — Dennett

The Non-Euclidean Ge­om­e­try That Wasn’t There

Defective Attitude Towards Suburbia

The Twin Deficit Phan­toms

Sleep Sync and Vertical Hold

More FUD In Your Eye

The Myth of Rub­ber­neck­ing

Keeping Intelligent Design Honest

Failure of the Amiga — Not Just Mis­man­age­ment

Maxwell’s Silver Hammer = Be My Honey Do?

End Unsecured Debt

The Digits of Pi Cannot Be Sequentially Generated By A Computer Program

Faster Is Better

Goals Can’t Be Avoided

Propped-Up Products

Ignoring ID Won’t Work

The Crabs and the Bucket

Communism As A Side Effect of the Transition To Capitalism

Google and Wikipedia, Revisited

National Geographic’s Obesity BS


Theodicy Is For Losers

Seattle Redux


Living Well

A Memory of Gateway

Is Apple’s Font Rendering Really Non-Pixel-Aware?

Humans Are Complexity, Not Choice

A Subtle Shift

Moralism — The Emperor’s New Success

Code Is Our Friend

The Edge of Religion

The Dark Side of Pixel-Aware Font Rendering

The Futility of DVD En­cryp­tion

ID Isn’t About Size or Speed

Blood-Curdling Screams

ID Venn Diagram

Rich and Good-Looking? Why Libertarianism Goes Nowhere

FUV — Fear, Uncertainty, and Vista

Malware Isn’t About Total Control

Howard = Second Com­ing?

Doomsday? Or Just Another Sunday

The Real Function of Wikipedia In A Google World

Objective-C Philosophy

Clarity From Cisco

2007 Macworld Keynote Prediction

FUZ — Fear, Uncertainty, and Zune

No Fear — The Most Important Thing About Intelligent Design

How About A Rational Theodicy

Napster and the Subscription Model

Intelligent Design — Introduction

The One Feature I Want To See In Apple’s Safari.