EDITORIAL
Rick Rosner
5139 Balboa Blvd. #303
Encino, CA 913163430
(818) 9869177
When I'm working in an especially scummy bar, I like to look around and wonder what would happen if the bar and its occupants were swept up in a Wizard of Oztype tornado/time warp and deposited in the wilderness to try to restart civilization.
Most of the time it looks pretty hopeless. Same for the occasional Mensa gettogethers I attended, looking for nerdybutattractive women. Most Mensa people didn't seem wellequipped to address the real world. (Meaning, by the standards I was using at the time, that they couldn't walk into a bar and get lucky.)
I guess I suspected the same of y'all, that somehow your IQ's made you less able, somehow, to deal with real life. But I'm changing my mind. The letters and other communication I've had with you are convincing me that you're competent, interesting, and adaptable. Some of you are defying the people who sneer, "If you're so smart, why aren't you rich?" by building successful careers. Langan, who drove some readers crazy, is showing himself to be personable and capable of drastic changes in style.
Maybe if we fell into a time warp, we wouldn't be able to rebuild civilization, but at least we should be able to build a lasting monument to our presence in the void.
PUZZLES AND LETTERS
Your editor and his bride went on a Holland America honeymoon cruise to the Caribbean and Bahamas. It was fun, the food was fancy and plentiful, and I only became slightly depressed that my level of comfort indicates that I now have no hope of ever accomplishing anything.
People go on cruises to drink, gain weight, gamble, buy overpriced bargains, and, if they're from Great Britain, to see the sun. For gamblers, the ship offered a simulated horse race, a mileage pool (where you bet on how far the ship will travel in 24 hours), bingo, a small casino, and Shipboard Lotto.
I spent a lot of time thinking about Shipboard Lotto because of the astoundingly bad odds it offers. I'll go ahead and tell you the odds, rather than have you waste your time plugging permutations into your calculators. My challenge to you is to come up with a game, real or imaginary, that is attractive to nonmathematicians and that offers lousier expected payoffs than Shipboard Lotto.
To play, passengers pick 15 numbers out of 75. (75 is convenient for the casino crew because they can use the 75ball bingo hopper to pick out the winning numbers.) And passengers played like crazy. A very conservative estimate would be that at least 1/4 of the ship's 1500 passengers spent $5 on a ticket every day. That means, among Holland America's four ships, at least half a million lotto entries annually.
Shipboard Lotto fliers proclaim, "Win up to $50,000." That is, people who correctly guess all 15 numbers win 50G's. (The small print says, "$50,000.00 limit each game aggregate payoff." I guess multiple winners would split the 50 thousand.) Here are the rest of the payoffs:
MATCH WIN
($)
0 50
1 5
6 5
7 20
8 250
9 1000
10 2000
11 4000
12 6000
13 8000
14 10000
15 50000
Here're the odds:
MATCH ONE IN MONEY RETURNED PER $
0 42.86 .2333
1 8.76 .1141
6 30.82 .0325
7 138.48 .0289
8 917.4 .0545
9 9099.3 .0220
10 139017 .0029
11 3425383 .00023
12 1.464 E8 <.00001
13 1.227 E10 <.00000015
14 2.533 E12 <.000000001
15 2.28 E15 <.000000000005
An average of less than 49 cents is returned for every dollar wagered, which is consistent with many state lotteries. The amazing thing is the probability of winning the grand prize. A passenger playing Shipboard Lotto once a day would have to take an average of over 300 trillion oneweek cruises to win the grand prize. That passenger would be riding the ms Westerdam for about 500 times the currently accepted age of the universe. I've never been to Las Vegas. Does keno offer odds that are this miserable?
Here's a very small chance to win some money back from me. Any subscriber who comes up with the next term in this series (by the end of June) will receive a refund of their $10.
3, 7, 19, 29, 71, 103, 103, 191, 233, 317, 577, . . .
(Yes, they're all primes.)
TRIAL TEST #2
Ronald K. Hoeflin
P.O. Box 539
New York, NY 10101
So far 22 people have tried Trial Test #1. This test, like that one, consists entirely of verbal analogies. Future tests in this series will include other types of problems. If you want to submit answers to this test and receive a score report, the scoring fee is $5.00, payable to "Ronald K. Hoeflin" at the above address. The entire projected series of ten tests is $25.00. There is one best answer for each analogy below. Feel free to use reference books of any kind, but do not consult other people directly.
1. Letter : Word :: Bit : ?
2. Too much : Hyper :: Too little
: ?
3. Stone walls : Prison :: Iron bars
: ?
4. Column : Row :: File : ?
5. Flower : Pansy :: Swindle : ?
6. Stayed home : 2 :: Had none : ?
7. Humbug : Bach :: Seek : ?
8. Coals : Newcastle :: Rough beast
: ?
9. European schism : Iron :: Asian
schism : ?
10. Enlightenment : Illuminati ::
Knowledge : ?
11. Mary : Jesus :: Necessity : ?
12. High : 5 :: Cloud : ?
13. Analog : Compass :: Digital : ?
14. Zee : Zed :: Billion : ?
15. Far : Moron :: Farther : ?
16. R2 : D2 :: THX : ?
17. Wise : Owl :: Dead : ?
18. Pride : Prejudice :: Sense : ?
19. Castor : Pollux :: Phobos : ?
20. Raker : Muck :: Hopper : ?
21. 38 : Pyongyang :: 49 : ?
22. U.S. : Death Valley :: The world
: ?
23. 4 : Mockingbird :: 2 : ?
24. Skull : Phrenology :: Face : ?
25. Ocean : Pellagic :: Ocean bottom
: ?
LETTERS
(some of which have been edited)
TO: MR. RICK ROSNER
I'M AN ASSOCIATE MEMBER OF T.O.P.S. (TOP ONE PERCENT SOCIETY). I REALLY DON'T WANT TO TAKE UP TOO MUCH OF YOUR TIME (THAT IS A LIE I DO WANT TO TAKE UP SOME OF YOUR TIME). I HAVE A FEW QUESTIONS I WOULD LIKE TO ASK YOU, AND THEY ARE AS FOLLOWS BELOW.
IF POSSIBLE COULD I AS AN ASSOCIATE MEMBER SUBSCRIBE TO NOESIS AND WOULD YOU SEND ME ALL THE INFORMATION ON THE JOURNAL. ALSO COULD YOU GIVE ME A FEW TIPS ON HOW TO DEVELOP MY MIND TO A MUCH HIGHER LEVEL. THREE MORE QUESTIONSI WOULD LIKE TO KNOW HOW COULD I GO ABOUT FREEING MY MIND FROM THE NONSENSE THAT MOST PEOPLE HAVE BEEN LED TO BELIEVE. ALSO I WOULD LIKE TO KNOW OF ANY BOOKS THAT YOU HAVE READ THAT WERE VERY HELPFUL IN YOUR INTELLECTUAL DEVELOPMENT. LAST QUESTIONI REMEMBER SEEING SOMEONE ON THE MORTON DOWNEY SHOW A COUPLE OF YEARS AGO, COULD YOU TELL ME IF IT WAS YOU?
WILL TRY NOT TO TAKE UP ANY MORE OF YOU TIME IN THE FUTURE, I KNOW YOU HAVE MUCH MORE IMPORTANT THINGS TO DO. ANY RESPONSE WILL BE GREATLY APPRECIATED.
REGARDS,
DONALD SCOTT
P.O. BOX 7080
GRAND CENTRAL STATION
NEW YORK, N.Y. 10163
Editor's comments: I've found I can appear much smarter, both to other people and myself, by pretending to be smarter, as if I were a method actor like Robert DeNiro getting ready to play Albert Einstein. When I pretend to be stupid, I find that I'm very successful at being a dope, and I experience similar success when pretending to be intelligent.
Observing other people helps me feel smarter, because people consistently do pathetically stupid things. (Of course, so do I.) Living in New York City is a fine opportunity to see people under pressure.
Getting into the habit of breaking down everyday life into mathematical terms is a way I've found to become better at math and better at dealing with everyday life. Have you ever studied statistics and probability? Many common beliefs don't stand up to statistical analysis.
If you read books at all, you've got an advantage over most Americans. I like to read trash, but carefully selected trash. My favorite book, and one that is hard to find but is available at the public library at 40th and Fifth, is Texas Celebrity Turkey Trot, by Peter Gent, an expro football player. It's all about rich, famous, powerful people who are assholes, and it convinced me that no matter how obnoxious I try to be, it isn't obnoxious enough. It changed my life. In general, I like books that show people being not very competent, but still salvaging their lives. These include:
First Light, by Charles Baxter
The Last Picture Show, by Larry
McMurtry
books by Scott Spencer (not including Last
Night at the Brain,
Thieves' Ball and Secret
Anniversaries)
books by Joseph Wambaugh from the 1970's
Panama, by Tom McGuane (Only because
the central
character, an outcast in his
hometown, puts a washtub
in his front yard and labels
it "Deranged Pervert
Wishing Well." He makes some money, too.)
Anyone can subscribe to this fine journal. It's ten bucks for six issues, payable to me. Yes, that was me on Morton Downey, trying to be obnoxious and not quite succeeding. If you become a subscriber, you will soon realize that you can take up my time, as I do not have anything important to do. Thanks for your letter.
HEYHOW ABOUT YOU ALL? HAVE ANY NOESIS READERS DISCOVERED TECHNIQUES FOR MAKING YOUR BRAINS WORK BETTER? HOW 'BOUT WAYS TO SEE THROUGH COMMONLYHELD NONSENSE? WHAT BOOKS AND MOVIES AND WHATEVER ELSE HAVE YOU FOUND HELPFUL OR ENTERTAINING? PLEASE, PLEASE WRITE (OR CALL) AND LET ME KNOW. I'LL PRINT IT UP AND MAYBE WE CAN BOOST EACH OTHERS' IQ'S, OR AT LEAST LOWER OUR BOREDOM Q'S.
Dear Rick:
I particularly enjoy your editorship for the interesting information you provide. Is there any other source for knowing how to detect a fake ID? I've never seen such information printed and find your more esoteric type of knowledge interesting.
While I'm at it, let me provide some feedback on your untitled puzzle on page 11 of the February Noesis. While I was able to obtain the answers to 22 of the 32 questions at first try, I am stuck on questions 12, 13, and 29. I would appreciate your publishing the answers.
Yet, my answer to #28 is so obscure that I wonder if it is correct: "101, a silly millimeter longer." That was an advertisement that was used for a year or so by a cigarette brand in the early 1960's to differentiate itself from the other brands during a period when companies were making the cigarettes longer and longer (starting from the approximate 80 mm length of the nonfiltered Camel and Lucky Strike).
Additionally, your last question, "96 T, by ?" strikes me as perhaps having a typo. If it was "76 T, by ?," then the answer would be "76 Trombones, by Meredith Wilson," which is the same genre as the previous question, "101 Dalmatians, by Walt Disney." If the question really is "96 T, by ?," then I'd appreciate the answer.
Cordially,
Robert D. Russell 3313 Circlewood Court Grapevine, TX 760516520
Editor's comments: I haven't seen much else in print about fake ID. Paladin Press, the book publishing branch of the Boulder, Colorado company that publishes the mercenary mag Soldier of Fortune, has put out one or two slim volumes on obtaining fake ID. Their intended readership is trying to get away with more serious stuff than underage drinking, and they recommend the "dead baby method," which involves finding the name and birthplace of someone born about the same time as the fake ID user, but who died during infancy. With a copy of the dead baby's birth certificate, everything else is possible (though getting a social security number has become more difficult). I've read in a couple places that, dumb as many criminals are, almost all serious felons have at least one alias and set of fake ID.
The company I work for, Grand American Fare, which owns a chain of 30 or so bars and restaurants across the southwest, has a training video on how to catch fake ID's. It doesn't go into much detail. A California company publishes a nationallyrecognized guide to stateissued ID's. The guide devotes less than a page to IDchecking technique. A U.S. government study, published in about 1976, says that ID fraud costs Americans about $10 billion annually. Fake ID shows up incidentally in various lowbudget movies and plays a more prominent role in a movie called Positive ID.
I've been disappointed in how little has been written and filmed about bouncers and fake ID. There was a British play called Bouncers, there was the pretty stupid Roadhouse, and one of F. Scott Fitzgerald's characters, Pat Hobby, consistently ran afoul of bouncers, but that's about it.
I agree with your answer to mystery Xerox test item 28, "a silly millimeter longer." Evidently, the mystery Xerox has been kicking around for a long time. This answer is consistent with what I believe to be the answer to item 29, "2 Mints in One." I don't know the answers items 12 and 13. I threw in "96 T, by ?" because the question mark looks like omitted information, but it's really the lead singer's stage name, at least according to late night TV ads for compilations of 60's music.
Dear Rick Rosner,
In response to your wife's remark that "These are brain people, can you really expect them to respond? Call me crazy, but I can't see Ron Hoeflin saying 'I had a little black cat named BooBoo Kitty'": I did in fact have a large black cat in my teens that I seem to recall that I called "Poopsie." I currently have a large yellow cat I call Big Boy and a small black cat I sometimes call Little Girl. My previous two cats I called Fluffy (19751990) and Toughy (19751985). I had a teddy bear in childhood but it was not exclusively mine and I don't recall ever having given it a name. However, our family members were given pet names.
My mother: Moke
My father: Daggedy, or Daggit
My sister: Gungit (the "ng" is pronounced
Myself: Ongit like the "ng" in "sing")
My brother: Blerp
My mother also called my sister "Spud" and when I was too young to properly pronounce my sister's name, Eugenia, my mother says I called her "Teeta" (probably an attempt to say "Sister").
I am righthanded.
I'd be curious to know the circumference of members' heads. Chris Harding once included this in a questionnaire for members of his "501 Society" (onein100,000 group). I thought it was a silly question at the time but subsequently noticed that my own hat size is extra large. (I never buy or wear hats.) My own head circumference seems to be roughly 23 1/4 inches. The psychologist Cattell, for whom various IQ tests are named, speculates in Eugenics Bulletin (the journal of the Mensa special interest group for eugenics) that blacks had to be tall and thin as a rule in order to radiate heat efficiently, just as Eskimos tend to be short and squat in order to retain heat. And the thinness would make it difficult for largeheaded black babies to be born, with a consequent negative impact on blacks' intelligence, since the pelvis of black women would be too narrow. Now that Caesarian births are possible, this negative impact on the birth of largeheaded black babies will no longer exist, Cattell points out.
I'm also interested in members' finger dexterity, although I see no easy way it can be measured by oneself. There is said to be a high correlation between finger dexterity and intelligence. I once took a finger dexterity test at an employment office operated by the State of California in Sacramento. The task was to pick up pegs out of a board, turn them upside down, and reinsert them in their holes, each end of the pegs being a different color for easy counting of how many one could turn over within the time limit. There were about 16 pegs per row and about 8 rows of pegs on the board. I could do an entire row of pegs more than the people sitting around me, which seemed to visibly disturb them, especially since we had to do the pegturning task 2 or 3 times in order to get our optimum performances, which gave them a chance to see that I consistently outperformed them.
Sincerely,
Ron Hoeflin
P.O. Box 539
New York, NY 10101
Editor's comments: My stepbrother, who is much hipper than me, took my name Richard and turned it into "Richnerd" and "Richturd". These degenerated into "The Nerd" and "The Turd." He didn't bestow his nicknaming gifts on any other siblings. During my periods of extremely obnoxious behavior, my stepfather would sarcastically refer to me in the third person as "Wonderboy" or, more simply, "Asshole". A bitter exgirlfriend called me "Dickless," after hearing the term in Ghostbusters. It didn't bother me, since, at the time, I was under the misapprehension that I am wellendowed.
Was Cattell in training to be a standup comedian? Richard Pryor has a similar routine on blacks' adaptation to the jungle, the punch line being, "If I'm so jungleadapted, why am I in Detroit?" Also, does Cattell believe in the cultural superiority of bigbrained Eskimos?
Mars Blackmon, a character from Spike Lee's movie She's Gotta Have It, would call Jesse Jackson "a chunky Chicken McNuggethead." Was Jesse Jackson delivered by Caesarian?
It seems to me that the white supremacists you see pictured at Klan rallies had trouble getting out of the birth canal. Maybe it's just bad haircuts, but Klan families tend to look like they were inexpertly delivered with forceps.
My skull has an average circumference, with a hat size of 7 1/4 or 7 3/8, but I have very big hair. Judging from my skill at masturbation, I have excellent finger dexterity. (I also used to be pretty good at delicate work on plastic models, making jewelry, and altering documents.) My large muscle coordination used to be awful, but now I can pass for a nonspaz if I absolutely must.
Dear Rick,
I can't resist responding to your question about stuffed animals. I can't remember having any attachment to stuffed animals, but I did have a collection of plastic dinosaurs, which kept me company during baths. I outgrew this, however, & got into the real thing. Not dinosaurs, obviously, but reptiles. I had a six foot water snake, which I would occasionally let swim in my 115 gallon aquarium. I had 12 aquariums & bred and raised fish. I was electrocuted by my electric catfish (close relative of the electric eel, which isn't really an eel, but a catfish). My water snake bit my hand once, & part of its fang was lodged in my left thumb for about 2 years. It was pretty disgusting. I also had a tarantula for several years. I used to dry my dead fish & collect them as ornaments (except for the real meaty ones, which would rot & cave in).
By the way, who was BooBoo Kitty? It's very familiar. 70's sitcom, wasn't it?
I read your "primes" stuff with interest. I'm really interested in number theory, too. There was a short time when I thought I would do good work in math, but I determined I was starting too late. That's not to discourage you, however.
Steve Sweeney
540 West Roscoe St., #482
Chicago, IL 60657
Editor's comment: BooHoo Kitty was a stuffed cat which belonged to Laverne on Laverne & Shirley. Way to spot discoera trivia!
Marshall Fox
30872 GA Tech Station
Atlanta, GA 30332
Dear Rick,
Thanks for mentioning me in last month's Noesis, and for dropping my name to Steve Sweeney. Steve wrote recently concerning Titan Test problem #33maximizing the number of pieces from a torus cut by three Mobius strips. He believes the true answer is higher than the one Ron Hoeflin is giving credit for, and after working on this awhile, I think the maximum is at least . One problem, I gather, is that the problem has been reworded since "trial test A" and has changed somewhat in meaning. In particularif I understand Steve's letterin the original problem the Mobius strips were confined to the interior of the torus, meaning that they had to pass through the centers of the perpendicular torus crosssections. The answer to this problem is . Lifting this constraint really adds two degrees of freedom, since now not only the placement of the pivots of the strips becomes important, but also the relative angles of the strips.
More please on metaprime sequences. In the meantime I'll see what I can do about confirming your conjecture for longer sequences, but my doctor has recommended that I give up computers and other carcinogenic substances.
My bio from Telicom is a little dated now (my life has changed so much since last October!) but in a nutshell I'm a 23yearold Math/Computer Science undergrad at GA Tech. I'll get out in about a year. I've been working odd quarters in the Radar and Instrumentation Lab, and this experience helped me land my first job offer (last week) at MIT Lincoln Labs. I'm excited about this, but I'm also a little apprehensive about moving to a city that doesn't even divide its streets and highways into lanes, and has toll booths outside private driveways. At least I'd probably eventually get to attend MIT grad school for free. I was also pretty nerdy in high school, but I outgrew it, as well as moved into an environment where the average student makes me look like Andrew Dice Clay.
Congratulations on your marriage, new job, and car wreck.
Marshall
Editor's selfserving babble: I know Ron Hoeflin doesn't like answers to his tests floating around, so in this and the following letters, I've mixed the answer to prob 33 with a few dummy answers. I hope this is a reasonable compromise.
What do you mean toll booths in driveways?
The Diceman is special to me since he played a bouncer in the John Hughes movie Pretty in Pink.
FROM STEVE SWEENEY TO MARSHALL FOX:
Dear Marshall:
I am a graduate student at the University of Chicago and I recently took Ron Hoeflin's Titan Test, for which I got a score of 40. I am writing you about problem 33. I'm pretty certain that you haven't taken the Titan Test, so I'll give you a wordforword transcription:
Consider the torus, a doughnutshaped solid that is perfectly circular at each perpendicular cross section, and a Mobius strip, which has a single 180degree twist and a uniform curvature throughout its length. Suppose a torus is sliced three times by a knife that each time precisely follows the path of such a Mobius strip. What is the maximum number of pieces that can result if the pieces are never moved from their original position?
The problem has been the source of a minor controversy as of late. I discovered that my answer was marked wrong when I got together with Rick Rosner in LA over Spring Break. Since he got a perfect score, that meant his answer of  was marked correct. But  is not the correct answer. It turns out that Ron Hoeflin wanted the Mobius strips to be confined to the interior of the torus, as they were on Trial Test A. But as worded in Omni, it becomes a much more challenging problem.
The reason I'm writing to you is because I'd be curious to hear what your answer would be. If you'd be interested, I'd like to see what you get. I realize that you may not want to spend the time to work on this problem, which is perfectly understandable. In this case, I'd like to send you my best solution (which is higher than my originally submitted answer) and see what you think. Perhaps you'd then see how to achieve to maximum.
One way or another, I hope to hear from you soon.
Sincerely, Steve
MARSHALL FOX'S REPLY TO STEVE SWEENEY:
Dear Steve,
The best answer I can come up with is .
This problem is much easier if the Mobius strips are confined to the interior of the torus, because the centers (i.e. the "pivot points") of the strips must lie at the center of any perpendicular crosssection of the torus. That is, if the pivot of any Mobius strip is offcenter in the crosssection, then, as the strip rotates, its edges will not meet the boundary of the torus. This really narrows the possibilities, since the only questions left are the initial angles of the strips, and what directions to rotate them. There are only two choices to the second, and if two strips rotate one way while the third rotates the other, then any initial set of angles will yield  pieces, as long as the angle between the first two is not zero.
But this is not what the Omni question asked, as knives are almost always longer than the things they cut. This is a HARD problem.
My solution occurs when the pivots are arranged in an equilateral triangle about the center of a crosssection, and two strips rotate one way while the third rotates the other, as before.
DIAGRAMS HAVE BEEN OMITTED DUE TO PUBLICATION FORMAT. SORRY.
It sounds to me from your letter that you may have gotten more than  pieces, and if so, I'd love to see how you did it. Incidentally, I'm pretty sure that the maximums for one and two slices are  and  pieces, respectively, so (if  is correct) there may be something very elegant at work here. Or maybe not. Doesn't the way the pieces weave around each other sorta remind you of a polytope turning itself insideout in 4space?
Thanks for asking me about thisI love this kind of stuff. I wish I liked school as much. I'm an undergrad at GA Tech. Interests include whitewater and backpacking. More on request.
Sincerely, Marshall
Editor's twaddle: As the French supergenius suggested in the previous issue, knowing a little about the author of an IQ test can help you score higher. My experience with the Mega was a huge advantage on the Titan, so when it came to ambiguous problems, such as the dissected torus, I was able to make glib simplifications and dance around ambiguities.
It took me awhile even to understand what was being asked on prob 33, and once I grasped it, I kept coming up with larger solutions until figuring out a way to get any number of pieces, even 979,428. (A wobbling, Parkinsonian knife does not necessarily have nonuniform helical curvature and can cut out all sorts of convoluted Mobius strips with uniform helical curvature.)
Back to the drawing board, where I made the problem as simple as possible (assuming a benevolent Ron Hoeflin), requiring each Mobius strip to pivot around the central point of each perpendicular cross section of the torus, which leads to a suspiciously straightforward solution, though one which was considered correct.
MAGIC
by Chris Cole
Magic is a form of puzzle that I have always enjoyed solving. By this I mean figuring out how magic tricks are done. Harry Anderson (of television's Night Court) was a wellknown professional magician  winner of the Society of American Magician's Magician of the Year Award  before he landed a TV series. He used to perform once a year at Caltech; I think he enjoyed mystifying the Nobel Laureates. I present below several of his best tricks. In the next issue, I'll give my hypotheses about how he did each trick. See how many you can solve, and if you can improve on my solutions. Also, I'd enjoy hearing about your favorite magic tricks.
THE INSUBSTANTIALITY OF TIME
The President of the Caltech Y (sponsor of Harry's performance) is asked by Harry to get the treasure chest that Harry mailed to him a week before the performance. He disappears and returns within five minutes with a small treasure chest (about 10" long by 5" deep by 6" high). He reports that the chest has been kept within the Caltech Y safe during the week; that this safe can only be opened by the President and a couple of trusted Y officers; that at no time during the past week has the chest been touched by Harry or anyone associated with Harry; and that he has just now retrieved the chest from where he put it in the safe a week ago.
Harry comments that most scientists will admit that the nature of time is a mystery. Newton postulated absolute time and space, but did not really believe it and Mach showed he was wrong. Einstein began to penetrate the true nature of time when he showed that time is relative. Modern quantum experiments show that time is not reversible. Mystics, who have used meditation to explore the nature of time for centuries, understand many things that modern science has not yet discovered. Harry will offer a small demonstration.
Harry is standing across the stage from the President. He asks for a volunteer from the audience. A local student volunteers and Harry asks him to unlock the safe with a large skeleton key that Harry produces. The volunteer approaches Harry, takes the key, crosses the stage, opens the chest with the key, removes a small rolledup scroll of paper, which contains the headline of that day's Los Angeles Times.
THE INSUBSTANTIALITY OF MATTER
Harry asks for class rings  the typical thick engraved brass kind with a large stone or emblem. Three people from the audience volunteer their rings. Harry enters the audience and gets the rings.
He reminds the audience that as scientists they know that the appearance of solidity is deceiving; that matter is mostly empty space; that even though quantum mechanics only touches the surface of the true nature of matter, it admits that matter can pass through matter, which is called "tunneling". Newton studied the ancient arts of alchemy secretly. Alchemists know that under certain circumstances it is possible for certain apparently solid substances, such as brass, to be passed through one another. Harry will offer a small demonstration.
He massages the three rings together in his hands. After a few seconds, Harry holds up three interlinked rings. He approaches the three volunteers, one at a time, in the audience. Each verifies that their ring is indeed interlinked in the chain. Harry again massages the rings, separating the rings one by one and returning them to their respective owners.
THE INSUBSTANTIALITY OF MIND
Harry asks for three volunteers. He asks the first volunteer to write down a three digit number, the second to write down a one digit number, and the third to find the Los Angeles white pages phone book. He collects the two numbers.
While waiting for the third volunteer to return, he reminds the audience that he has demonstrated that they have a lot to learn about time and matter, and that the same holds true of mind. Even though many people in the socalled "hard" sciences are skeptical, research has shown that many people have extrasensory perception. By rigorous ascetic discipline, Harry has honed his own latent abilities to a fine edge. Harry will offer a small demonstration.
The third volunteer returns with the phone book. Harry hands him the two slips of paper, instructs him to turn to the page indicated by the three digit number, look down the page to the line indicated by the one digit number, and concentrate on the name on that line. After a few seconds of concentration, Harry correctly reads off the entire line.
THE HARDING MULTIMAX
I'm always looking for ways to sleaze myself a few extra IQ points, so I was happy to see this in the May, 1991 InGenius:
[from Ron Hoeflin:] This test was designed by Chris Harding, founder of the ISPE, the world's first successful 99.9 percentile highIQ society. He claims that this tenitem test has a ceiling of 211 IQ based on 16 IQ points per standard deviation. But . . . , this would correspond to a rarity of one 501,128,385,303. However, simply by guessing one has one chance in 9,765,625 of getting a perfect score. Thus, one is about 51,000 times more likely to get a perfect score by guessing than by actually solving the problems. So unless I have overlooked something, this test would appear to be severely flawed psychometrically.
Zow! Just ten questions with a shot at breaking IQ 200, plus no math! I jumped right in, psychometric flaws be damned. While I was wondering whether I should print the test in Noesis, Chris Harding sent me a letter requesting me to do so. Here it is.
The questions that follow were designed to reveal how clearly you are able to think. There is one correct answer in each case. Circle the letter corresponding to the one you have selected as your answer.
(1) The party political
system arises as a result of:
a. the containment of diversity in a single expression of ambition.
b. external forces set up by internal needs.
c. a natural response to apparent and similar opposites.
d. natural factor associations in people's traits.
e. the attempt to deceive and coerce.
(2) The concept of
infinite intelligence is:
a. false because it is nonrelatable to finites.
b. false because the function is an emergent property.
c. meaningless on the human plane.
d. beset with difficulties of a philosophical nature.
e. undefinable in logicomathematical terms.
(3) Love is:
a. a concern for another's wellbeing.
b. positive transference.
c. an expression of the spirit.
d. a learnt function.
e. a process of object assembly.
(4) Consciousness and
selfidentity are:
a. the highest functions known to the highest states of mind.
b. a function in part of the cloudy states of intelligence.
c. an echo function of obsolete intellectual responses as they die out.
d. derived from precise nonerror randomness.
e. a useful illusion born of a form of external internalization.
(5) Art is definable as:
a. the composition of incomplete expressions.
b. a form of problem solving in the social domain.
c. expressions of beauty as seen in the eye of the beholder.
d. an upmarket form of the seek [sic] for community consensus.
e. embodiment of the attributes of emotion and judgement.
(6) Most people see
imprisonment of criminals:
a. as a way to find the means to rehabilitate them since they must then comply
with certain requirements.
b. as a means to teach them useful skills that may result in less destructive
modes of behavior.
c. as a means to disable those who are seen as all too able.
d. as a means of keeping society as safe as possible.
e. as a way to rid society of its dangerous element.
(7) The final outcome of
increasing world population will be:
a. everyone on earth starving to death.
b. disease on a scale never seen before in world history.
c. a decrease in the complexity of living systems.
d. exhaustion of all support systems and natural resources.
e. wars between countries as each blames its neighbors for its own
difficulties.
(8) Issues about the
existence or nonexistence of God are:
a. deeply rooted in the historicocultural context of which they are a
function.
b. aspects of the hypersocial matrix at points of non intersection of its
equations.
c. the approach to unresolved areas of marginal awareness.
d. elements of the defense resolution aspect of our lives.
e. the process of struggle as it becomes revealed to us.
(9) Happiness:
a. is at base an economic question for the most part, though having other
elements to it.
b. is an issue that can only be resolved through questions about interpersonal
dynamics.
c. depends upon one's upbringing and current mental health.
d. depends upon one's situation at the time and how one relates to these factors.
e. means gains occurring in relation to factors like surprise, etc.
(10) Genius is:
a. a fiction of society and an incorporation of its mythology.
b. social approbation of the highest kind.
c. a declaration of the spirit of man.
d. merely an ascendancy of the highest values.
e. brought by the age.
THANK YOU! for participating in ISPE's testing program
Office use only:
Points________Q________P________T________
Test serial number_________________
You may return your completed test to the author,
Dr. Christopher P.
Harding
P.O. Box 5271
Rockhampton Mail Centre
Queensland 4702
AUSTRALIA
MORE METASEQUENCES
I received quite a bit of stuff about metasequences, both through the mail and, from Dean Inada, electronically. Chris Cole, who thinks that this journal could function as a clearinghouse for readers' meanest problems, suggested that I publish the electronic correspondence between Dean and me to show how much information and advice we can get from each other. I agree, the only problem being that I can't really operate a computer, so much of our correspondence is illedited or lost. The tattered remains comprise part of this article. Told of this, Dean asked me to avoid printing any of his statements which turn out to be "lies," his words for intuitions that turn out to be not 100% correct, but I've found most of his comments to be very accurate and perceptive.
In my communication with Dean, I state my belief that periodic repetition of terms (a as a factor in every other term, z in every 103rd term, etc.) is the key to rich metasequencesones that generate plentiful alternate sequences. However, I've found classes of rich aperiodic metasequences that have fractaltype structures. Such metasequences are formed by removing primes from the standard sequence S and replacing them with integer metaprimes. The simplest such aperiodic metasequence is S2,+4, where multiples of 2 are eliminated, but multiples of 4 are included. It goes 3,4,5,7,9,11,12,13,15,16,17,19,20,21,23,25,27,28,29,31,33,35,36,37,39,41, . . . Numbers in S missing from S2,+4 have a fractal distribution that gives S2,+4 a density approaching 2/3. Primes eliminated from S can be combined to form metaprimes, as in S3,5,+15. (This metasequence would have a density approaching (2/3)(4/5)(15/14) = 4/7.)
Thank you for all your responses!
FROM CHRIS LANGAN: (This Langan stuff is rather rough metaphysical sleddingless metaphysical stuff follows.)
I have a hunch you're right about the greater pugnacity of east coast barflies. Still, there's quite a bit of traffic between New York and Los Angeles, and even the Saturday Night Fever crowd may soon be going intercoastal. Beware the man with sloping shoulders and a B'ooklyn accent.
Regarding your interesting article MetaPrime Sequences, I'm not quite sure how to interpret your premise. but at a superficial glance, it looks like there isn't any choice between b and a2. That is, a^n = a + 1 for no positive values of a and n. Yet, each element in a "counting" sequence must be derived from the last one by mathematical induction . . . by adding 1. This is a pretty tight constraint. Some believe it to be the reason that there is only one permissible counting sequence. If other possible sequences exist, they require a redefinition of mathematical induction . . . and thus of unity itself. This will involve more than just skipping certain primes, or introducing nonintegers into the sequence. To preserve the arithmetical regularity that makes N useful, the redefinition would have to be regular as well. But our only concept of regularity seems to be based on N itself, and on the extended algebraic symmetries defined in terms of the natural numbers. so it almost looks like no matter what "other" sequences exist, they'll have to mirror N up to isomorphism, or be nonnumerical and useless from a human standpoint. That would seem to confirm your hypothesis re the optimality (or "richness") of N: it's optimal to us because it defines our ultimate logicomathematical syntax.
Of course, the inductive constraint just given is merely the ultimate reduction of something you've stated yourself: that every nth term must be a multiple of n. I've merely reduced it to something like, every term must be a multiple of one. Because you're admitting fractional (and even irrational) values for letters in order to define an alternate sequence N', this constraint is superficially violated. On the other hand, skipping certain primes yields only a proper subset of N, which merely lacks corresponding ramifications of the Ntree and therefore cannot be "as rich," given that the frequency of primes converges downwardly on a limit (1/(log x)). Either way, any attempt to restore mathematical induction to N' ultimately entails isomorphism to N. So from a Cantorian settheoretical standpoint, as long as you want to talk about "counting", you're still talking about N (or aleph). In other words, the sequence of counting numbers is an inevitable tautology of the human cognitive syntax.
The number 1 is the identity of the multiplicative group of N. Thus defined, it seems to be in some respects arbitrary. But no matter what such an identity specifically represents, it must be apprehended as a unity. So we must use the identity of N to make sense of it. Since this identity stands for N itself, N is an unavoidable tautology of whatever analysis we undertake concerning the identity of N' under multiplication, and so N' itself.
Let's put it another way. Say you skip 2, resulting in the sequence (1,)3,5,7,9, . . .= a,b,c,a2, . . . . Instead of 2, 3 is now the second number. I.e., skipping the cardinal number 2 has placed the cardinal number 3 in ordinal position 2. It is still every third number that is a multiple of 3. But the third cardinal number is now 5! Cardinality and ordinality are now out of synch, and we no longer have a "counting" sequence.
Yet, your premiseas far as I understand itseems to be not only original, but highly conducive to numerical insight. Any perspective that helps us "stand outside" our cognitive syntax is of order so high as to be almost transcendental. You are trying to get a glimpse of the "genesis algorithm." I could write more on your approach to number theory, but let me get this in the mail before Chris Cole gets left with the wrong impression about his claim that he has "invalidated" the CTMU. [This is an excerpt from a longer letterEditor]
[an excerpt from a later letter from Chris Langan]
In my remarks on Metaprime Sequences, a^n = a + 1 is said to hold for no positive values of a and n. Naturally, that should have read "for no positive integer values of a and n" . . . i. e., for no elements of the standard counting sequence. One must still ask how to express a^n after the prime integer n has been dropped from the standard sequence. The only obvious answer is to continue to use the standard sequence for such expressions even after substituting a nonstandard sequence at the object level. That's what I meant when I wrote that we have to use N to make sense of N'. But the objectlevel variations still seem full of potential insight.
Later,
Chris L.
Editor's comments: Those of you who wrote about metasequences tend to want to make them behaveDon't they haveta have a prime between n and 2n, between n and n^2? Well, like little Popeyes, they are what they are, and they can misbehave in various ways. Those that misbehave too badly, however, end up poor. Bad behavior includes noninteger primes (except for logarithmicallyexpanded sequences where each prime is taken to the same noninteger powerS^(1/2) = root 2, root 3, root 4, root 5, . . . . It works the same as the standard sequence.) and knocking out a divergent infinity of primes.
I started thinking about metasequences because I wanted a 20" neck that would be wider than my ears. At the gym, another guy and I would put towels over each other's faces and try to wrench each other's heads floorward. The other guy was a chain smoker and just wanted to shake the phlegm out of his throat. He was also a professional mathematician who was working to simplify the computer proof of the four color theorem. We'd silently count out the number of repetitions we were giving and receiving, and we always disagreed by 10%.
That two guys who were good at math couldn't count to ten together made me wonder what the deal was. Eventually, I had to conclude that overfamiliarity with numbers made at least one of us a bad counterthat one of us had developed a streamlined semiconscious counting system that contemptuously skated over certain numbers.
Thus, I tend not to believe, as Chris suggests, that "the sequence of counting numbers is an inevitable tautology of the human cognitive syntax."
I do agree that it's kind of silly to skip numbers and not skip the corresponding powers of those numbers, as must new subscriber Steve Sweeney:
FROM STEVE SWEENEY:
Regarding your prime stuffyou seem to have left out some significant things. For example, what about aa, bb, etc.? Seems you should have aa between a and ab. [I doEd] Another approach is to forget about the standard primes all together. If a,b,c can be arbitrary, why do you need a2, a3, etc.? A more familiar representation of the standard sequence would, I think, go as follows: a,b,aa,c,ab,d,aaa,bb,ac,e,aab, . . . . Now you can ask why aaa comes before bb, for example. Does it have to? I'm not saying your approach isn't interesting. But if you don't want to be tied down to the conventional prime sequence, why assume it in coming up with coefficients?
What's the "best" way to get the following sequence: a,b,c,d,e,f,g,h,aa,i,j,k,l,ab,m,n,o,p,q,ac,r,s,t,ad(=bb),u, . . . ? It's the composite numbers broken down into new "primes". Now 4 is the first prime, 6 the second, 8 and 9 the third and fourth, etc. The first "composite" number is 4x4 = 16. What if we repeated the process and threw out all these new primes? Now the first prime is 16, the second 24, the third 32, etc., with the first composite number being 16x16. Do we get any kind of pattern now (say in terms of the relative number of primes separating the composites)? My guess is no. But what if we take the limit of the process? Do we approach any pattern?
What's the limit of this sequence of metasequences? (I.e., do we approach a definite pattern in terms of the "composites"aa,ab,ac,bband the relative distance between them?)
1.
a,b,(aa),c,(ab),d,(aaa),(bb), . . .
2 3
4 5 6
7 8 9
2. a, b, c, d, e, f, g, h,
(aa), i, j,
k, l, (ab), m,
4
6 8 9 10 12 14 15
16 18 20 21 22 24 25
n, o,
p, q, (ac), r,
s, t, (bb=ad), u, v, (ae), . . .
26 27 28 30 32 33 34 35
36 38 39 40
3. a, b,
c, d, e,...
(aa),...(ab),... (ac),...(bb=ad),..(ae),...
16 24 32 36 40 256
384 512 576
640
4. a, b,... (aa),.
..(ab),.. .(ac),... (bb=ad),...
256 384 65536
98304 131072 147456
*
*
*
A pattern seems to form. What does this tell us? Maybe nothing. But I think if we want to find a pattern in the primes, it may come in some limiting process, such as the one described above.
As you can tell, I'm a numberjunkie, too. I like playing around with this stuff and, in response to your final sentence (in the Noesis article), I am interested in seeing more.
I'd better get off this stuff now before I get hooked. So good luck to you & Carole, & let me hear from you.
Steve
Editor's comment: Years ago, I tried to mess around with metasequences that used noninteger exponents and got nowherelogarithmically spotty shotgun patterns in nspace. It was just too hard for me. Maybe that's where Langan's necessary equivalence between cardinal and ordinal numbers kicks in. I dunno. To keep things on a level I can deal with, I've gotta use the unadulterated standard sequence as exponents. Maybe one of you can take a shot at what happens when certain integer exponents are disallowed.
ELECTRONIC MAIL FROM DEAN INADA
Date: Fri, 19 Apr 91 19:12:48 pdt
From: dmi (Dean Inada)
To: chris, rsterman
Subject: Re: noesis 59
I'd appreciate it if you'd take a look at the prime article & tell me
whether it's clear and easily amenable to computer attack.
Thanks,
Well, I think I was able to comprehend what you were saying.
An arbitrary metasequence of the type you describe sounds equivalent
to the sequence of positive latice points swept by a tilted plane in
an infinite dimentional lattice.
Which might be a known problem.
I would expect that programming a computer attack would not
be too hard, but there may be a combinatorial explosion
at around 20 terms or so...
[For primes, the slope of the plane would be (a*ln(2)+b*ln(3)+c*ln(5)+...)=z]
Date: Sun, 21 Apr 91 02:03:14 pdt
From: dmi (Dean Inada)
To: rsterman
Subject: Re: noesis 59
Status: OR
said. Where should I look to see what work has been done on it?
I don't know.
A vague deja vu of the problem in its tilted plane form is
all I can think of.
In two dimentions, it might come up in computer graphics related work,
or in rational number theory. The general counting problem sounds
like combinatorics, but I can't think of anything more specific,
sorry.
Date: Wed, 8 May 91 23:55:57 pdt
From: dmi (Dean Inada)
To: rsterman
I just recieved a letter from Steve Sweeney, who has been discussing the
torus and mobius strips problem with Ron, and is seeking support for
his answer of at least 24 given the wording on the Titain test.
He may be intersted in publishing his thoughts on the problem in noesis.
Also, after seeing the meta prime sequence article in print,
I'm not sure I understand how you are measuring richness when
you claim that the standard sequence is the richest branch.
Date: Sun, 12 May 91 12:34:27 pdt
From: dmi (Dean Inada)
To: rsterman
Subject: Re: noesis, programming
Chris, as you suggested, I'd like to run Dean's responses to
my query about metasequences, but I don't know how to retrieve letters
Hmm, I don't know if they'd be suitable for publication.
I might be careless about lies and misspellings and not well thought out
remarks in a dialog, where feedback and correction is easier.
e.g. I think what I said about the relationship of partition counting
to your problem was a lie. (It may be closer if you replace "meta sequences
of n terms" with "sequences with n prime powers" or something like that,
but you'd also have to fudge the "without regard to order" part too)
Dean, what I mean in talking about the richness of the main
sequence versus other sequences is that in a decision tree, paths can
be traced from the origin through many forks. Some forks contain few
choices, as at a normal intersectionsome forks contain many choices,
as at the Arc de Triomphe in Paris. I think that the aggregate of the
By "aggregate" do you mean "sum"?
number of choices seen at each point in the main sequence must in some
By "point" do you mean "after n terms of the sequence"?
way be larger than the aggregate of the number of choices along any
other branch. At each point in the main sequence, you're choosing
from an average of the natural log of the number of decisions you've
already made (after 20 forks, you're choosing from an average of 3
alternatives at each forkafter 50, anaverage of 4 alternatives).
Other branches give you an average of fewer alternatives at each fork.
There are some restictions of possible sequences for primes, e.g.
there is always another prime between p and 2*p, which implies e.g.
that you will never see something like "a a^2 b" in the sequence.
Should other meta sequences have this same restriction?
There are other, much deeper restrictions, none of which are covered
by my plane slicing model, which could make the enumeration problem
much trickyer.
But ignoring such things, considering the two extreme sequences,
a a^2 a^3 a^4 ..., and a b c d e f ... those have only two branches at each
point, which does suggest that the optimum is somewhere in the middle.
Primes come on average about every n*ln(n), how does the sequence
generated by haveing them occur exactly every n*ln(n) compare with
the real sequence?
How about "nearby" sequences such as (n+ln(ln(n)))*ln(n)?
From: rsterman
Subject: reply to Dean
To: dmi., chris (CHRIS Cole)
Date: Mon, 13 May 91 2:05:13 PDT
Cc: dmi (Dean Inada)
Dean
aggregate is sum, each point is after n terms, as you asked.
The plane slicing model is equivalent as long as the slices are
parallel for each metasequence.
Closetothecenter sequences based on a prime every n/ln n
terms should be nearly as rich as the standard sequence. The richest
sequences preserve as much periodic behavior of terms as possible.
That is, the standard sequence, which I believe to be richest, is
arranged so that the ratios between successive terms approach 1the
ratios are 2/3, 3/4, 4/5, etc. Differences in the ratios of succesive
terms become increasingly small. The richest sequences minimize
repetition of ratios between successive termsin the sequence 1, a,
b, a^2, the b is thrown in to break the ratio a/a^2 into two less
abrupt ratios. That is, the sequence 1/a, a/a^2 contains two equally
abrupt ratiosb buffers the ratio a/a^2.
As counting numbers get larger, the ratios of succesive terms
become less abrupt. Such decreasing abruptness involves periodic
appearances of termsthat is, a shows up in every other term, b in
every third term, c in every fifth.
Sequences that are less rich are less periodicthey're
missing terms from the standard sequence, so that clunky ratios show
up periodically, like missing fenceposts2,3,4,6,7,8,9,11. Missing
primes are missed opportunities to have a smooth sequence. The larger
the omitted prime, the less frequent the missing terms, the richer the
metasequence.
I'm sure this isn't very clear. It all becomes clear with a
few examples I'm not equipped to give at 1:20 in the morning after a
miserable shift in a bar. However, I'll continue stumbling along.
Call the standard sequence S. The sequence S2, with all
multiples of 2 omitted, is less rich than the sequence S37, with
every 37th number omitted. Sx sequences remain very periodic and
very rich in their sum of alternate paths. Obviously, every Sx
sequence with a finite number of omitted x's has a basic period
containing a pattern of omitted terms that endlessly repeats. S2,3,5
has a period of 30. It goes(1,)7,11,13,15,17,19,23,29, 31,37,41,43,
. . . Even with all the gaps, its still nicely periodic. Even with
all the initial primes in a row, it eventually gets into composite
numbers and ends up fractionally as rich as S.
Some metasequences aren't even fractionally as rich as S. S
plus any non whole number ends up being 0% as rich as S. For
instance, S+2.52,2.5,3,4,5,6,6.25,7,7.5, . . . turns real ugly.
It's not missing fenceposts, it has fenceposts increasinly close
together, making for a tattered, misaligned sequence.
In the sliced nspace model, different sequences are
represented by different vectors perpendicular to the slices. Rich
sequences are represented by fat vectors (like searchlights)the
vector can wobble a litttle without changing the sequence. Poor
metasequences have vectors that are more constrained. The partial
sphere of possible vectors (where each point on the partial sphere
represents a different metasequence) consists of two different types
of points representing the infinitely constrained poor metasequences
and the less constrained rich sequences.
To amplify a previous pointrich sequences are Sx where x is
any finite combination of primes, such asS2,+43,4,5,7,9,11,12, .
. . (2 is left out, but 4 is a metaprime)
S3,5,7,+152,4,8,11,13,15,16, . . .
Such sequences are fractionally as rich as S. S2 is
probably half as rich, S5, 4/5 as rich, S101, 100/101 as rich.
There are semirich, semiperiodic metasequences. These are S
minus an infinite number of primes such that the product of (p/(p1))
for each omitted prime remains finite. S  the 2nd, 4th, 8th, 16th,
32nd, 64th, etc primes is an example (I think). It is S3,7,19,53,127,
. . . (Maybe it would diverge. How about S11,101,1st prime>1000, 1st
bigger than 10,000, . . .)
Thanks for slogging through this.
Rick
Date: Mon, 13 May 91 15:26:31 pdt
From: dmi (Dean Inada)
To: rsterman
Subject: Re: prime clarification
terms should be nearly as rich as the standard sequence. The richest
sequences preserve as much periodic behavior of terms as possible.
Is this an empirical observation, a provable/proven theorem,
a belief, or a definition?
That is, the standard sequence, which I believe to be richest, is
arranged so that the ratios between successive terms approach 1the
ratios are 2/3, 3/4, 4/5, etc. Differences in the ratios of succesive
terms become increasingly small. The richest sequences minimize
This should be true of many sequences.
repetition of ratios between successive termsin the sequence 1, a,
b, a^2, the b is thrown in to break the ratio a/a^2 into two less
abrupt ratios. That is, the sequence 1/a, a/a^2 contains two equally
abrupt ratiosb buffers the ratio a/a^2.
There must always be a prime between p and p^2.
(do you require meta sequences to have this property?)
I'm sure this isn't very clear. It all becomes clear with a
few examples I'm not equipped to give at 1:20 in the morning after a
miserable shift in a bar. However, I'll continue stumbling along.
I hope you don't feel obligated to answer all my questions immediately.
One of the nice features of email is that it can be read/replied to
at ones convenience.
Some metasequences aren't even fractionally as rich as S. S
plus any non whole number ends up being 0% as rich as S. For
instance, S+2.52,2.5,3,4,5,6,6.25,7,7.5, . . . turns real ugly.
It's not missing fenceposts, it has fenceposts increasinly close
together, making for a tattered, misaligned sequence.
I don't see the connection between richness and periodicity.
Clearly the terms generated by primes in the rational integers
will be maximaly smooth. But I thought richness refered to the
number of ways of continuing a sequence given only the factorizations,
here, a, b, c,a^2, ab, ac,b^2, d, bc, ...
\ \ \ \ \ \ \ \ \
a^2 a^2 d d d d bc e e
    
b^2 a^3 a^3 a^3 a^3

c^2
For an agreggate branching of 23 up to this point.
(2.5 could have as easily been anywhere from 2.45 to 2.66
as far as we can tell from the sequence)
In the sliced nspace model, different sequences are
represented by different vectors perpendicular to the slices. Rich
sequences are represented by fat vectors (like searchlights)the
vector can wobble a litttle without changing the sequence. Poor
metasequences have vectors that are more constrained. The partial
sphere of possible vectors (where each point on the partial sphere
represents a different metasequence) consists of two different types
of points representing the infinitely constrained poor metasequences
and the less constrained rich sequences.
You mean the rich sequences have the most nearestneighbors on the sphere?
to periodic appearance of terms tends to generate a "poor"
metasequence. Primes showing up on a strict schedule would force
previous terms to reappear aperiodically, but I'll attempt a
Again, it is not obvious to me why this should be so.
(How about primes in a field other than the rational integers?
e.g. the field of numbers of the form a+b*sqrt(3),
(some of these fields are not Unique Factorization Domains,
so prime may not be the same as irreducible. In general,
p is prime when p divides a*b implies either p divides a or p divides b))