We're playing a game in which you, the player, start with a number sequence. There is a rule governing which number comes next, and whoever determines the rule will recieve $10. Any one can play, but I tagged the people who i think will be most interested.
If you guess a number, I will tell you if it is correct, and if so, I will add it to the existing sequence. Please only guess one number each day. Please only guess one number at a time, dont try and fill in a section of the sequence.
If you guess the rule, I will tell you if you are correct or incorrect. If correct, you win $10. If incorrect, you may not guess the rule again for 3 days.
Original sequence:
2, 4, 6
The sequence so far is:
2, 4, 6, 10, 18, 30, 50, 82, 134, 218, 354, 622, 623, 630 47 comments Updated about a month ago
Craig Fleischman (Indiana) wrote at 7:44pm on July 13th, 2007 10? Message - Delete
Dan Margolis (Japan) wrote at 8:31pm on July 13th, 2007 7 Message - Delete
Jeff Borack wrote at 1:03am on July 14th, 2007 10 yes, 7 no Delete
Dan Margolis (Japan) wrote at 9:52am on July 14th, 2007 Its like fibonacci sequence except starting at 2. The next digit is the sum of the two previous digits. So it would be 2, 4, 6, 10, 16, 26, 42, 68, 110...
So... X0 = 2, X1 = 4, Xn = (Xn-1 + Xn-2) Message - Delete
Jeff Borack wrote at 12:16pm on July 14th, 2007 Incorrect Delete
Dan Margolis (Japan) wrote at 12:38pm on July 14th, 2007 Worth a shot...I can't deduce much from so few numbers... Message - Delete
Elliot Alyeshmerni wrote at 7:06pm on July 14th, 2007 im gonna go with 18 Message - Delete
Jeff Borack wrote at 8:24pm on July 14th, 2007 a job well done Delete
Yvette Monachino wrote at 8:08pm on July 15th, 2007 30 Message - Delete
Jeff Borack wrote at 10:47am on July 16th, 2007 30 works Delete
Yvette Monachino wrote at 11:06am on July 16th, 2007 50 Message - Delete
Jeff Borack wrote at 11:35am on July 16th, 2007 good Delete
Elliot Alyeshmerni wrote at 2:25pm on July 16th, 2007 82, still havent gotten the sequence down so this is a bit of a guess Message - Delete
Jeff Borack wrote at 2:33pm on July 16th, 2007 good Delete
Elliot Alyeshmerni wrote at 3:19pm on July 16th, 2007 i think we all got this sequence now.. Message - Delete
Jeff Borack wrote at 3:36pm on July 16th, 2007 i dont think anyone has it. but i welcome you to guess. If your right, $10. If your wrong, at least you'll save yvette! Good luck. Delete
Peter Dahlke wrote at 7:31pm on July 16th, 2007 134 next? Message - Delete
Jeff Borack wrote at 8:07pm on July 16th, 2007 yup Delete
Elliot Alyeshmerni wrote at 10:49pm on July 16th, 2007 218 Message - Delete
Jeff Borack wrote at 11:15pm on July 16th, 2007 218 Delete
Victor Baranowski wrote at 10:16am on July 17th, 2007 IDK where it started, but assuming we started with 2, 4, 6 the sequence is:
Xn = X (n-1) + [(X(n-1) - X(n-2)) + (X(n-2)-X(n-3))]
or something like that... Message - Delete
Jeff Borack wrote at 10:26am on July 17th, 2007 Interesting guess, I thought people were gonna say Xn = X(n-1)+X(n-2)+2, but both are wrong. Sorry Vic. The more interesting question is: why did it take so long for someone to guess? Is the reward for guessing the correct answer to low or is the penalty to high? Delete
Jeff Borack wrote at 10:45am on July 17th, 2007 I'm changing the rule of 1 rule guess/week. You can now guess once every three days. Numbers are still once a day even though elliot broke that rule and i accepted the number. Delete
Elliot Alyeshmerni wrote at 11:43am on July 17th, 2007 this a answer works for every number except 6 and 18, but i'll put it down anyway
X(n)=2X(n-1)-X(n-3) Message - Delete
Victor Baranowski wrote at 12:22pm on July 17th, 2007 Ya, that was similar to mine. Why the sequence goes from 10 to 18 is the tricky part of this whole thing, which makes me think the equation is going to be pretty ugly or wierd... maybe jeff made a mistake :P Message - Delete
Victor Baranowski wrote at 12:23pm on July 17th, 2007 Oh, and I might as well guess 354... Message - Delete
Jeff Borack wrote at 2:19pm on July 17th, 2007 a) the solution is beutiful b) i didn't make any mistakes yet c) 354 is good Delete
Victor Baranowski wrote at 2:46pm on July 17th, 2007 Can I cite a) in response to your b) ? Message - Delete
Jeff Borack wrote at 8:20pm on July 17th, 2007 Hmmmm, I'm not sure. It depends on when you think the mistake was made. Technically it did come before b), but i could also argue that the mistake what made when i clicked the "Add your comment" button.
a) the solution is... very nice and good b) i didn't make any mistakes in the number sequence yet. c) web browsers and AIM should have spell checkers. this isn't the 20th century anymore. Delete
Tait Kowalski wrote at 3:48pm on July 18th, 2007 Sequence goes x(n) = x(n-1)+2*x(n-3)
so the next number = 354 + 2*134 = 622
next number is 622 Message - Delete
Jeff Borack wrote at 4:41pm on July 18th, 2007 Welcome Tait! That is the wrong rule, but ill accept your guess at the next number. Delete
Elliot Alyeshmerni wrote at 6:28pm on July 19th, 2007 the next number is fuck you jeff, just give us the answer lol Message - Delete
Jeff Borack wrote at 6:47pm on July 19th, 2007 Sorry elliot, want me to call the Whaaaaaaaaaaaaaaaambulance? Delete
Victor Baranowski wrote at 4:29pm on July 22nd, 2007 is the next number 620? Message - Delete
Jeff Borack wrote at 7:23pm on July 22nd, 2007 hmm strange guess. 620 is not a number Delete
Victor Baranowski wrote at 8:46am on July 23rd, 2007 howabout 623? Message - Delete
Jeff Borack wrote at 12:40pm on July 23rd, 2007 : ) 623 is the next number Delete
Craig Fleischman (Indiana) wrote at 12:55pm on July 23rd, 2007 630? Message - Delete
Jeff Borack wrote at 12:59pm on July 23rd, 2007 630 is good Delete
Victor Baranowski wrote at 1:51pm on July 23rd, 2007 Solution: the next number is whatever number is guessed, as long as it is higher than the previously guessed number. Message - Delete
Jeff Borack wrote at 2:28pm on July 23rd, 2007 hahaha, yup. it took a lot of time but not a lot of guesses. i expected the guessing to to into the hundreds of thousands. do you accept paypal? Delete
Victor Baranowski wrote at 2:35pm on July 23rd, 2007 no, i accept shots and beers the next time we hang out. Message - Delete
Yvette Monachino wrote at 4:10pm on July 27th, 2007 that is the dumbest sequence i have ever heard of Message - Delete
Jeff Borack wrote at 5:08pm on July 27th, 2007 It's about thinking outside the box, yvette, something i wouldnt expect most MATH majors to understand! : p Victory for the engineers!!! Delete
Yvette Monachino wrote at 2:08pm on July 30th, 2007 aw thats a cute remark, knowing that you don't actually know what real math is i won't take that as an insult, and the only victory you accomplished is adding yourself to the long list of pompous engineers, so congrats :) Message - Delete
Jeff Borack wrote at 2:52pm on July 30th, 2007 While I might be pompous, I unfortunately can't be considered much of an engineer. I did bioengineering, which certainly doesnt count, and i've never actually engineered anything. Neither has vic, hes in law school.
It is true that i don't know what real math is (although i would love for you to teach me). However, I would imagine that real math does involve thinking outside the box on occasion. In this particular example, it required you to test a number you thought was not part of the sequence. If you believed you had found the sequence, and contintued to test numbers that fit that sequence, you would never derive the answer. By simply testing a number that does not appear to fall into the sequence, such as 2 million, it's easy to find the solution.
Does this sound like any 'real' math problems you have ever encountered?
I am teaching a class, and I write upon the blackboard three numbers: 2-4-6. “I am thinking of a rule,” I say, “which governs sequences of three numbers. The sequence 2-4-6, as it so happens, obeys this rule. Each of you will find, on your desk, a pile of index cards. Write down a sequence of three numbers on a card, and I’ll mark it ‘Yes’ for fits the rule, or ‘No’ for not fitting the rule. Then you can write down another set of three numbers and ask whether it fits again, and so on. When you’re confident that you know the rule, write down the rule on a card. You can test as many triplets as you like.”
Here’s the record of one student’s guesses:
At this point the student wrote down their guess at the rule. What do you think the rule is? Would you have wanted to test another triplet, and if so, what would it be? Take a moment to think before continuing.
The challenge above is based on a classic experiment due to Peter Wason, the 2-4-6 task. Although subjects given this task typically expressed high confidence in their guesses, only 21% of the subjects successfully guessed the experimenter’s real rule, and replications since then have continued to show success rates of around 20%.
The study was called “On the failure to eliminate hypotheses in a conceptual task.” Subjects who attempt the 2-4-6 task usually try to generate positive examples, rather than negative examples—they apply the hypothetical rule to generate a representative instance, and see if it is labeled “Yes.”
Thus, someone who forms the hypothesis “numbers increasing by two” will test the triplet 8-10-12, hear that it fits, and confidently announce the rule. Someone who forms the hypothesis X-2X-3X will test the triplet 3-6-9, discover that it fits, and then announce that rule.
In every case the actual rule is the same: the three numbers must be in ascending order.
But to discover this, you would have to generate triplets that shouldn’t fit, such as 20-23-26, and see if they are labeled “No.” Which people tend not to do, in this experiment. In some cases, subjects devise, “test,” and announce rules far more complicated than the actual answer.
This cognitive phenomenon is usually lumped in with “confirmation bias.” However, it seems to me that the phenomenon of trying to test positive rather than negative examples, ought to be distinguished from the phenomenon of trying to preserve the belief you started with. “Positive bias” is sometimes used as a synonym for “confirmation bias,” and fits this particular flaw much better.
It once seemed that phlogiston theory could explain a flame going out in an enclosed box (the air became saturated with phlogiston and no more could be released). But phlogiston theory could just as well have explained the flame not going out. To notice this, you have to search for negative examples instead of positive examples, look into zero instead of one; which goes against the grain of what experiment has shown to be human instinct.
For by instinct, we human beings only live in half the world.
One may be lectured on positive bias for days, and yet overlook it in-the-moment. Positive bias is not something we do as a matter of logic, or even as a matter of emotional attachment. The 2-4-6 task is “cold,” logical, not affectively “hot.” And yet the mistake is sub-verbal, on the level of imagery, of instinctive reactions. Because the problem doesn’t arise from following a deliberate rule that says “Only think about positive examples,” it can’t be solved just by knowing verbally that “We ought to think about both positive and negative examples.” Which example automatically pops into your head? You have to learn, wordlessly, to zag instead of zig. You have to learn to flinch toward the zero, instead of away from it.
I have been writing for quite some time now on the notion that the strength of a hypothesis is what it can’t explain, not what it can—if you are equally good at explaining any outcome, you have zero knowledge. So to spot an explanation that isn’t helpful, it’s not enough to think of what it does explain very well—you also have to search for results it couldn’t explain, and this is the true strength of the theory.
So I said all this, and then I challenged the usefulness of “emergence” as a concept. One commenter cited superconductivity and ferromagnetism as examples of emergence. I replied that non-superconductivity and non-ferromagnetism were also examples of emergence, which was the problem. But be it far from me to criticize the commenter! Despite having read extensively on “confirmation bias,” I didn’t spot the “gotcha” in the 2-4-6 task the first time I read about it. It’s a subverbal blink-reaction that has to be retrained. I’m still working on it myself.
So much of a rationalist’s skill is below the level of words. It makes for challenging work in trying to convey the Art through words. People will agree with you, but then, in the next sentence, do something subdeliberative that goes in the opposite direction. Not that I’m complaining! A major reason I’m writing this is to observe what my words haven’t conveyed.
Are you searching for positive examples of positive bias right now, or sparing a fraction of your search on what positive bias should lead you to not see? Did you look toward light or darkness?