Combinatorial Explosion: Theory Is Worse Than You Thought

Combinatorial Explosion: Theory Is Worse Than You Thought

A lot of kids have alphabet blocks. Twenty six wooden squares with the letters of the alphabet on them.

One way to arrange them is just like in the song, “A B C D E F G…” You know the one. You’re humming it in your mind even now. But there are other ways to arrange the blocks, too. Like “B A C D E F G…” And “C A B D E F G…”

Now set your kid, or grand kid, or an NPR listener, down and have them make every possible arrangement that can be made with the blocks. As a guess—don’t do any math—how long do you think this might take? Suppose it takes one full second to make each arrangement.

Ten minutes? A hour? Maybe a full Saturday afternoon?

About 12 or 13 quintillion years, depending on snack and bathroom breaks. A quintillion is 10^18 years, or 1,000,000,000,000,000,000 years. Some say the universe is about 14 billion years old. That means it would take the lifetimes of a bit under 1 billion universes, each lasting 14 billion years, to make every possible arrangement.

All sizes are relative, and this time, relative to times in our experience, at least in this universe, is big.

You can make this big a bit smaller by speeding the process up. We did one arrangement each second. But why not two? Or even ten? Ten is speedy, yet not hard for a computer. So we’ll imagine our kid is “AI”. A brute beast computer algorithm, one that is much faster than an ordinary child. Let’s do a million arrangements each second. Literal computer child’s play! Now how long will it take for all arrangements?

The math is easy. A million is 6 zeros. All we have to do is subtract those 6 zeros from the 19 we had for one-per-second. That leaves us with 13. So 10,000,000,000,000, or ten trillion years. Still longer than the age of the universe, and by a fair measure.

What if we did a billion arrangements per second? That’s 9 zeros; subtracting gives us 10, which is ten billion years. That’s nicely in the range of the age of the universe.

So, if we did one billion-with-a-B arrangements each second, it would take about the lifetime of the universe to get through them all.

Now let’s add a twist. Suppose we take all these alphabet blocks and dump them in a big bin, and we shake it around for some time, and then dump the lot on the ground. Further suppose there is a machine with 26 square holes, into which fit one block per hole.

This machine is constructed to flash a green light if the blocks when in each of the holes line up as in the alphabet song, and it flashes a red light otherwise. With me?

Our next step is to get a Harvard Studies graduate and have her try various combinations to try to get the light to glow green. She won’t have any familiarity with the alphabet song, which is white supremacist—the letters on the block will only appear to her to be strange scratchings—so she won’t know what order to insert the blocks. She will instead have to try various combinations “at random” until she finds the one that works.

What is the probability that, on her first try, she gets the light to glow green? Which is to say, that she gets the right order? Well, I won’t bother you with the calculation, but it turns out to be about 2.5×10^-27. Which, if you like to stare at long numbers, is 0.0000000000000000000000000025.


What is the probability she gets it right on her second guess? That might be trickier, until we recall this is a Harvard Studies graduate. Meaning, we suppose, there is no possibility she will remember the order she tried first. So the probability is, again, 2.5×10^-27.

Even if she could remember, there is no clue in the sequence she tried first that she was “close” if it was wrong. For instance, suppose she got the first 24 letters right, and ended her first try “…U V W X Z Y”. The light glows red. She won’t know that all she had to do is swap the last two.

We already saw that it would take her a mighty long time to try every sequence, even if she could keep track of her previous guesses. Too long, even though it turns out she might not have to try every sequence until she got lucky. But whatever. She will never make it.

If we let the alphabet blocks be certain chemicals, the Harvard Studies graduate standing in for “random” arrangements of them, the machine’s green light indicating genes that function, then we have, in rough form, the neo-Darwinian theory of “random” mutations causing evolution.

Now don’t screw your fedora around your ears and start making hersterical noises like that mother on the phone in a Christmas Story. The analogy is, indeed, inexact, as all analogies are.

But it errs on the side of simplicity. Real so-called random mutations are vastly more complex, and there have been, the claim runs, untold trillions upon trillions upon trillions upon of them. You get the idea. Our block example represents just one new mutation of a dirt-simple gene.

We also see that the “random” theory cannot be believed. But don’t worry! Some kind of intelligent design, as long as we’re careful defining these words, must be true.

If you doubt any of this, try it yourself. Make it easy on yourself and only use the letters “A B C D E F G”—G for gene, you see.

I got this idea from a post by Ann Barnhardt.

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  1. McChuck

    Add in the life span and breeding age of the individual members of the species, multiplied by the number of generations it takes to fix the mutation across enough members to matter, and once again, there simply isn’t enough time. Even if you parallel process this across millions of members of a species – which would require even longer to fix a mutation as dominant.

  2. john b()

    My ear worm is “Twinkle Twinkle Little Star”

  3. A through G? Seven letters give a mere 7! = 5040 permutations; at one per second a clever puzzler could run the board in an hour and 24 minutes Of course, adding one more letter (Holy Hellfire Horse Hockey!) multiplies all this by 8. Do the arithmetic for a few more add-ons, and you’ll understand the enormity of combinatorial growth. Donald Knuth wrote a thoughtful book riffing on all this, Things a Computer Scientist Rarely Talks About.

  4. bob sykes

    Except, of course, Darwin’s Theory of Natural Selection is NOT about randomness, it is about selection. The combinatorial argument is simply irrelevant to the theory.

  5. Incitadus

    The Big Bang Theory may be more of a theological accommodation than an
    actual fact. The universe could well be hundreds of billions of years old. There
    is an intelligence operating at biological levels of functioning alien to our own.
    Pretty much all our vaunted cognition and theory is capable of is tossing a monkey
    wrench into it from time to time with varying results.

  6. Steve

    Close, but no cigar, @bob sykes.

    His explanation relies on both adaptive radiation and natural selection. His fixation with finch beaks is all about adaptive radiation. It just doesn’t work unless you start running through the various combinations to get different outcomes.

  7. Nonsense. And, more importantly, not relevant to the question of intelligent design.

    Please read “speculations” on my site – it starts like this:

    ” In his recent Claremont essay: “Giving up Darwin”, Dr. David Gelernter of Yale University’s Computer Sciences faculty argues that Darwin’s theory of evolution is like Newton’s understanding of motion: simple, elegant, and widely accepted but inadequate beyond a minor subset of the phenomena it purports to describe.”

    The basic error here is that the alphabet machine does not, in reality, accept all blocks into each hole. Instead only those blocks meeting very strict conditions fit (number holes 1 –>n; blocks fitting into hole K must meet conditions determined by holes 1–>k-1). So for two holes k, k1 with k>some small x, the number of possible pairs (i,.e. pairs the machine will accept) is very small.

  8. Johnno

    THE SCIENCE ™ demands, DEMANDS, that EVERY sequence shalt begin with the letters L – G – B – T – D – I – E

    That is the first rule of modern THE SCIENCE ™ club.

    Minority blocks FIRST! Then DIE! And damn the consequences!

    If nothing is working, it is obviously the colonialist green/red light machinery that needs to be changed and backed by the coercive force of law! Tear it down! Then redistribute non-functional Green lights for everybody until Equity is achieved! We will all be equal when we are all atomized and frozen in space!

    The basic error here is that the alphabet machine does not, in reality, accept all blocks into each hole. Instead only those blocks meeting very strict conditions fit (number holes 1 –>n; blocks fitting into hole K must meet conditions determined by holes 1–>k-1). So for two holes k, k1 with k>some small x, the number of possible pairs (i,.e. pairs the machine will accept) is very small.

    A smaller statistically impossible probability is still a statistically impossible probability.

  9. Cary D Cotterman

    The alternative being that an old guy wearing a sheet, with a big white beard and the voice of James Earl Jones (with a slight echo), just decided to invent it all one week.

  10. Briggs


    The green light is selecton.

  11. John M

    Billions of years old is based on the circular theory that carbon dating is calibrated to ice layer dating while ice layer dating is calibrated to carbon dating.

    Darwin didn’t write about Finches. See Lack, D. L. 1947. Darwin’s Finches. New York, Cambridge University Press.

    Prehistoric fossils that still have viable genetic material can’t be over 100,000 years old. DNA breaks down by then.

    There are numerous articles of earth age, genetics, intelligent design, and other studies not beholden to Darwin at

  12. The True Nolan

    @bob sykes: “Except, of course, Darwin’s Theory of Natural Selection is NOT about randomness, it is about selection. The combinatorial argument is simply irrelevant to the theory.”

    Selection among what? Among varying offspring of the prior generation? What is it that made those different offspring vary from their parent(s)? Was that variation random, or was it guided in some manner?

    The combinatorial argument is very relevant to deciding how that variation occurred.

  13. Brad Tittle

    This being one of the last bastions of sanity. Anyone seeing a sharp uptick in Flat Earth crap… I have flat earth coming at me left and right.

  14. C-Marie

    Very interesting!!! Of course, we all know that God is the Creator, and He allows that which He allows!!

    God bless, C-Marie

  15. Johnno

    The alternative being that an old guy wearing a sheet, with a big white beard and the voice of James Earl Jones (with a slight echo), just decided to invent it all one week.

    Yes, exactly! I’m glad that you understand very well that this is the only option you have left to you, just as you know very well that all such artistically licensed depictions of the old guy in a sheet with an echoing James Earl Jones voice also didn’t happen randomly either. These creations are all well documented and told to us directly by their authors.

  16. Stack in Prague

    Impressive. Only a little trouble in that: it is totally wrong.

    It seems Feynman’s definition of the difference between math and physics applies to molecular genetics as well.

    There are only 20 letters in the protein alphabet. A bit less than in the English one. There are some 20400 protein coding genes in the human genome. Protein size varies greatly, but say it is 100 amino acids in average. That gives 20400*10^40 combinations. Quite a lot of time to try them all – therefore creator proved?

    But wait! What about that pesky genetic code? That one that stores instruction regarding the sequence of amino acids in proteins. It is triplets of four letters. Codons they call them. There are 61 codons for 20 amino acids. So some amino acids have more codons. Now, my lord Star Statistician, how many combinations are there in the genetic code for the unique sequence on an average protein? You know, that number is needed as a proof of nonexistence of evolution. Before you start combinatorics, two additional constrains: any combination having anyone of three stop codons anywhere but at the end of the sequence stops at at that codon, and (a simple one) only one certain amino acid is allowed at the beginning of all combinations. Now show your calculation.

    One question for an expert about intelligent designer: could Darwin meet Him when he (Darwin) noticed the differences among those apparently recently created birdies on the Galapagos?

  17. Rudolph Harrier

    I dunno if evolution through random mutation and natural selection is what happened or not, but every conversation about it online has convinced me that it’s an article of faith for atheists.

  18. Stack in Prague

    “… an article of faith for atheist.” Yet another example of oxymoron.

  19. Rudolph Harrier

    Because atheists are smart boys who only ever use logic and reason to determine everything they know?

    That in itself is another belief that atheists take on faith. In fact that is one of the big appeals of atheism for the duller parts of the population; you get to pretend that you are smart simply because you are an atheist regardless of what you actually say.

    What I’ve noticed the pushback from atheists to any doubts relating to evolution exceeds their pushback to anything else, including statements that God created the universe or that Christ rose from the dead and later ascended into Heaven. And you always get statements from the likes of Bob Sykes that misunderstand how things are even supposed to work in evolution in the first place. It’s okay if “evolution” actively selects traits, meaning bypassing random variation entirely, because then “evolution” works, i.e. not “God.”

  20. Yeah, except IRL you don’t get only two signals, green and red, you get tons of signals which means the poor Hardvard graduate is going to get a pretty strong hint it was very close in that off-by-one case. 🙂

    Or else you’ll be willing to explain to us how “AI training” works? Random mutations and selection, as far as the eye sees…

  21. Patient

    Right-wing disinformation, to discredit the fundamental patriotic contribution to the theory of evolution and the greatness of folk socialist ortodox genetic science in the context of the cancellation of Russian culture, of course. But this is normal in the unenlightened world of American hegemony. Things are very different in the Multipolar, where even every Russian first-grader knows that:

    Updated: 28.01.2024

    The main contribution of S.S. Chetverikov is to prove the connection between genetics and the theory of evolution. In his work, he showed that a population is a special level of organization of the living world, at which elementary evolutionary phenomena are manifested. Chetverikov also proved that mutational phenomena occur continuously in nature, which accumulate in populations and are a hidden reserve of hereditary variation.

    In the course of the scientist’s research, it was found that the struggle for existence is clearly manifested in living nature, while the arena for it is ecosystems, and the basic unit of evolution is populations.

    Thus, thanks to the efforts of S.S. Chetverikov and other scientists in the 1920s and 1930s, a comprehensive genetic-ecological approach to the study of the theory of evolution was laid, thanks to which it was possible to overcome the crisis situation in evolutionary science and create the basis for the formation of a synthetic theory of evolution, or modern Darwinism.

  22. Patient

    And, do you know the case of Kudryavtsev?

    Director of the Institute of General Genetics of the Russian Academy of Sciences Alexander Mikhailovich Kudryavtsev said that people in the past lived 900 years

    The head of the Russian Academy of Sciences Commission on Combating Pseudoscience, Academician of the Russian Academy of Sciences Evgeny Alexandrov, 28.03.2023: “These are sick people”

    Etc., big party. Yesterday I read news that they finally removed the villain from the post (Kudryavtsev I mean, not the boss of the fighters with “pseudoscience”. To make it clear what science is and what are some minor pseudoscientific religious views in the only country in the world where the president personally and openly states the assertion of Christian values.)

  23. john

    “Faro shuffle”

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