The most boring number in the world is...
scientificamerican.com
scientificamerican.com
My brain just did not want to accept "7" as the fourth number in the series.
I even drew it, and it was still damned difficult to make my hand offset the line to leave that shit piece in the middle!
Makes me wonder if there's some forth dimension that humans are quantized to ignore. Like right now, there is an alien traffic jam on some sprawling inter-dimensional superhighway caused by a human who plopped down like a duck to check on the price of an NFT.
Make the offset bigger and then the middle piece can be plenty big!
Interesting that your mind went to dimensionality - because the way to get 8 instead of "7" is to cut in 3D!
Do a + on the top, and then cut horizontally along one of the layers of icing! Voila! 8 cake pieces :D
On a more serious (ish) note, can you cut a 3D pie in 4D space? If I go with a 2D circle in 3D space, by default you can't cut it down the extra dimension. But what if the circle is curved in 3D space, say by attempting to wrap it part the way around a sphere. Even if the circle was originally in the XY plane, you can now perform a Z plane cut which would allow going from 4 to 8 pieces. So ideally if you were to warp the 3D pie around a hypersphere, you would have a way of cutting it to make 16 pieces.
Now I'm imagining a conspiracy among economists: to only ever show us top-down pie charts while funneling all the depth into their own slush fund. :)
You just have to be careful not to have two lines with similar slope, or you'll require a very large pie!
Lazy Caterer make Grug angry. Grug smash stupid caterer in stupid face.
Dividing a shape means "divide fairly", because that's what you've done thousands of times. Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.
Math / what's possible doesn't care one bit about that though. It takes work to train yourself to move past other training.
> Dividing a shape means "divide fairly", because that's what you've done thousands of times.
There are fundamental game-theoretic reasons for a fair split, yes. But there are also, arguably, more "pure" aspects at play, unrelated to agent dynamics. For example, out of the possible division ratios, there is an infinite number of them that give you two different pieces of cake, but only one ratio that gives you two indistinguishable pieces. What it means, among other things, is that[0] the 50/50 split is a singular one[1] that removes one bit of information from the "cake split into two pieces" system! Any other split requires you to track which piece is which (if only to be able to tell whether I just took the smaller or the larger one); the 50/50 one does not (the pieces are the same, it's pointless to ask which one I ate).
> Drawing some lines means "in an aesthetically pleasing way" because that's what you've seen a billion times. Etc.
The cake example above involves a special solution that removes bits from the problem. I feel the same is the case with a lot of aesthetics - we seem to have a preference for patterns that compress well. I don't think it's a peculiarity specific to human minds - it seems to me to be fundamental enough that all minds would naturally end up with a bias towards well-compressible pattern.
--
[0] - Take this piece of cake with a grain of salt, because I'm just a smartass, not an actual mathematician.
[1] - Ignoring the degenerate cases, i.e. the 0/100 and 100/0 splits.
1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386...
I was ready to attempt drawing that, but then I saw the other possible definition:
> Also number of regions in 4-space formed by n-1 hyperplanes.
So much easier to visualize it like this!
I wonder how much of this is due to a cultural sense of fairness and what proper pie cutting looks like.
A friend mentioned being at a party once where the guest of honor cut the first slice for themself and they chose to make a glancing slice that maximized their surface frosting to volume ratio.
I wonder if it is possible to make just the right cut so all pieces are "equal" (in area)?
Is there a mathematical tool to make experiments like this graphically / visually? I define three cuts as lines that are offset by "x" from the center of the circle and as I vary the value of x from zero to r, the areas of each cut piece should be computed and plotted.
> My brain just did not want to accept "7" as the fourth number in the series.
The sequence that you get from the difference of successive numbers in the list is rather predictable: 1,2,3,4,5,6,7, ...
1, 2, 4, 8, 16, 31
2^0
2^1
2^2
2^3
2^4
2^5 ... -1 ?https://en.m.wikipedia.org/wiki/File:PancakeCutThrice.agr.jp...
> So the split into interesting and boring numbers seems to stem from the judgments we make, such as attaching importance to prime numbers.
Given the context of the rest of the article, about what numbers appear in OEIS[1] sequences, I'd argue that the author has assumed the conclusion from the start. If you look only at numbers that people have deemed interesting, you....only find numbers that people have deemed interesting.
There's a much deeper (maybe-philosophical) question left unasked about the nature and structure of numbers. I was hoping the author would at least wave to that.
[0] https://en.wikipedia.org/wiki/Interesting_number_paradox
I have only a handwavy notion of it, based on some on-line articles (IIRC one of them might have been a blogpost or a paper by Scott Aaronson), but the general idea is that the things we find interesting exist in the middle between low and high-entropy states.
For example, take matches. A box of matches is relatively low-entropy: they all sit there neatly arranged in parallel rows, pointing the same direction. Matches in a box are boring. Now, dump the box onto the table. The matches are now arranged in a random pattern. This is a high-entropy state, and it's even more boring than the low-entropy one. Now, start putting those matches into some pattern - suddenly, things get interesting. One of the particularly interesting configurations would be a self-supporting 3D structure made of those matches - and that's, in terms of entropy, about half-way between matches in a box, and matches on a pile.
Put in information-theoretic terms, it seems that we find both highly predictable and highly unpredictable configurations boring: the former likely because there is just not much new information to pay attention to - but the latter, seemingly because it's too unpredictable, and we round it up to "pure noise".
So my question is, is there any formal description of this concept? A mathematical answer to why we find both order and disorder boring, and are most interested in things in-between? Some kind of natural quantity that tracks -E² (with E being entropy), and which would be fundamentally useful to maximize?
If you can find it, please share the blog or paper where you found the idea.
So relative to this given formal language, you can have a smaller integer that doesn't have a compact representation, doesn't have representation "smaller" than itself (and so might be labeled "boring"). And you can refer to the smallest of these integers in some meta-language but you can't refer to this smallest integer within the given formal language (and still have the language be consistent).
Related to: Kolmogorov complexity, Godel's Second Incompleteness Theorem, Chaitin's Number Omega
Anyway, it'd be cool if the OEIS had the numbers up to some limit. Maybe 2^16 so it fits into two bytes. It would be at most 1.2MB per entry, but generally way way smaller.
I agree! Another opportunity for a paper though!
If f(180) = y = 20067, what does the graph of f(x) look like?
How quickly would f(x) grow? Linearly? Quadratic? Exponential? Logarithmic? Any bets?
Interestingly, this will create a paradox because it could itself be expressed as an OEIS sequence, which would mean that it does not belong in the sequence... but if it doesn't belong, then it should belong(Russell's paradox anyone :D).
Even if you have an algorithm to compute them. I appreciate your optimism thought. But OEIS is made by pretty smart people, if it was so simple, it would have been made.
It would be great to make it when possible but the bias will remains.
I don't believe this to be true. Maybe the people who make OEIS have been satisfied with having the description and a few examples.
It would only work well for strictly increasing sequences. I don't know of any sequence that increases slowly and is difficult to compute. If it increases quickly, then you'll quickly pass the limit. If it's not increasing, then you'd have to prove which numbers are under the limit and that could be very challenging in some cases.
As far as I understand it, though, these B-files aren’t indexed by the search engine.
This isn’t actually correct. 1729 is the smallest number expressible as a sum of two positive cubes in two different ways. The smallest number expressible as a sum of two cubes in two different ways is 91 (3^3+4^3 and -5^3+6^3).
+832375400086541 if you can provide a proof through mathematical induction.
So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.
Sure there is, lex order on Re and Im and then inherit from the reals.
That can't be true either since -91 = (-3)^3 + (-4)^3 and -91 = 5^3 + (-6)^3. Since -91 < 91, the statement has to be false.
In fact, it is not the number x with smallest value |x| that can be expressed as the sum of two cubes in two different ways since, 0 = (-1)^3 + 1^3 and 0 = (-2)^3 + 2^3.
It appears that 91 is the smallest positive integer that can be expressed as a sum of two cubes in two different ways.
There is no value to this comment. I mostly just wanted to be annoying. I hope I'm right, at least.
There is always value in truthful statements. You are right that 91 is not the smallest number as was implied by the parent. -91 is. After all it was the parent who brought in the negative integers into the mix.
As long as we're not talking about nonzero value.
I think GP made a worthwhile point, but in general the universe is full of uninteresting truths. Like take two large, distinct numbers, assert that they are unequal. Is that specific truth valuable?
Since annoying provides negative value, your statement is incorrect.
Also, there is a catalog of interesting numbers: the Penguin Dictionary of Curious and Interesting numbers. A college prof of mine would read an entry before each class. https://www.thriftbooks.com/w/the-penguin-dictionary-of-curi...
[0] https://jeremykun.com/2011/07/28/false-proof-twenty-word/
The interaction and his name stuck in my mind, even as I entered an era of underachievement. Today he has a handful of degrees, including a PhD/DPhil in Mathematics at Oxford. He has no recollection of or interest in our short meeting that remains lodged in my memory.
> 20,067
Because it’s the smallest number not in any of the sequences they describe. It wasn’t worth the ride.
edit: I should add that all of this was well known to Ramanujan, with whom I share a birthday. He famously was able to refute the uninterestingness of 1729.
X
X XX
XX XX
XXX XX
XXXXXXXX
1234567890More than that. Every number is either even or odd! Super common attributes don't do enough.
> I found this number interesting enough to spend a few minutes on it, which is more than I give most numbers!
I would say you didn't find the number interesting, you found the context interesting. There are billions of similar numbers that could have gone in the same slot and you would have spent the same effort on them. But if you were set free to look for interesting numbers, without being fed this one, you never would have glanced at it. And more specifically, this is a context that would basically never produce the same number again; it's not a sequence with a generator.
Half of them try this keysmash method, and half of them try to come up with numbers without such randomness.
Then we have them each vote for a number they find interesting.
In that case, numbers like the above will get one vote. Lots of other numbers in these sequences will have tens, hundreds, thousands of votes.
Are the votes wrong? Is your number really the most interesting? Is it even the most interesting number between n-10 and n+10?
Throwing out half the resumes you get won't help you pick a candidate with better luck. Happening to randomly generate a number once doesn't make it interesting.
Nothing in this discussion is specific to that number. All the interestingness of this discussion should be split trillions of ways across all the similar keysmash numbers.
PROOF BY CONTRADICTION: If there are boring numbers, then there must be a minimum absolute boring one; which makes it an interesting number. QED
Therefore, there is an infinite amount of numbers that are not interesting for any other reason than that they are not interesting.
The fact that the most obviously not interesting number in the infinite pile of uninteresting numbers is interesting because it is so obviously uninteresting does not distract from the infinite pile of not interesting numbers below it.
We can assign that uninteresting number a value of 1/1.
If the second most obviously least interesting number is interesting because it is the second least interesting number, then we can say that it has an interest value of 1/2 of the most obviously least interesting number.
The next one would get 1/3, and the next 1/4, and so on and so on through 1/infinity, which converges to mean that there is a total of 2 points of interest in the uninteresting group. (The first is that there is a most least interesting number, and the second in that there is an infinite pile of uninteresting numbers aside from that).
I'm gonna do some hand waving here and say that this means that there is a point where the number are so large and incomprehensible to human or machine minds, where the values of the numbers are so entirely lost in computational morass that to assign the nomenclature that differentiates that number from its brethren requires more computational power than there will ever be atoms in the universe capable of being used for computation between the beginning and end of time.
Somewhere out past that is a number whose 1/"interest" value is so denominatorily close to infinity that it is, in fact, the least interesting number even when taking into account that the least interesting number is interesting for its uninterestingness. Remember, we are already working within the infinite set of fundamentally uninteresting numbers.
Therefore, there is at least one uninteresting number that is completely boring and uninteresting despite the fact that uninteresting numbers are interesting because of their uninterestingness.
That number must be the most uninteresting number regardless of whether it is interesting because of its lack of interestingness because there has already been an infinite number of interestingly uninteresting numbers to the point where there is no interest left to account for its uninterestingness.
This number would be so uninteresting as to be impossible to even think of. The moment you can conceptualize a number it would automatically be excluded from the list of candidates for the most perfectly uninteresting number. Even the concept that this response has generated has created a false concept of the least interesting number. The least interesting number is beyond this explanation and the abstract concept of a "number" that this conveys, the number that you could say would be interesting because of its uninterestingness, is not the number that is being expressed here. It is too uninteresting to be identified as anything concrete, too elusive to be conceptualized, too mystic to be grasps, too meta to be connected to the tangible universe, and there are so many of them you could hold every infinite uninteresting number that may or may not possibly exist in the aplm of your hand and you still wouldn't be able to grasp it because of how uninteresting it is, even though the description of how difficult the number is to express how uninteresting the number is is interesting, the number itself means nothing either directly or by comparison.
{ a∈ℤ | ∀S∈O a∉S }
Then you need to import a full copy of that OIES (as definitions should be self-contained as much as possible). That importing is what would stop you -- the definition of the problem would end up far too huge.