||The optimal compression scheme|
||(which is, for all practical purposes, impossible
Suppose you'd like to invent a new compression scheme
for pictures taken by people with their digital cameras.
We want this compression scheme to be good through at least
the year 2050 (at which point it will be replaced by
JPEG 2050 :)
How small can you make
those files? Maybe 50% smaller than traditional
JPEG? Or 80% smaller? Well, here's a (purely theoretic)
way to store every
file in less than 8 bytes.
estimated that the world population by 2050 will be
just under 1010. Imagine that every single person
in the world took a digital photograph every minute of every day
from now until 2050. How many pictures would that be?
A slight overestimate is 1010*60*24*366*50 <
2.64×1017. If we had 58 bits, we could store
258 > 2.88×1017 different integers,
and 8 bytes = 64 bits is more than enough.
So how does the compression work? Easy: we just number every
digital picture taken between now and the end of 2050. This is
picture 1, this is picture 2,... , this is picture 20,392,394,910.
The compression algorithm simply stores which picture number was
taken. The decompression algorithm reconstructs the picture.
Of course there is a lot "cheating" going on there. First,
there is of course no practical way to build such a decompression
algorithm, since it would need to "know" all of the pictures which
would be taken between now and 2050 -- certainly predicting the
future is a harder problem than image compression! Second, even
if we could build such a decompression function, this function
would effectively be storing all the data of all the pictures ever
taken in this time period! That's a huge function, dude. And
having to store that function would almost obviate the benefits
of any compression scheme. Finally, the compression algorithm itself
is also practically immpossible to write, since we have no way to
really know which "picture number" we are currently at -- but that
point is moot next to the first two.
In any case, I still think we can come up with a better image
compression scheme than the current popular formats! Clever
coders, consider this a challenge :)
||A brief note about people who have only taken the Staten Island ferry once in their lives|
||They're still there.
||The Evil that is Google Print|
||Ignorance has held back many a good
thing. Luckily, we have responsible
journalism to hold back that pesky
ignorance. Until, that is, those
writers feel threatened in their
Suppose someone proposed a new tax
law in which everyone except authors/journalists/publishers
got a huge tax cut. Nothing would get
worse for those media minions; they'd
just miss out on the cut.
Do you think it would pass?
To give a yet more obvious thought experiment:
suppose someone wanted to undertake
an operation to own a copy of virtually
every book ever made, and implement a search
mechanism by which virtually anyone could
find the book they wanted,
and then read that book for free.
Do you think authors would like that?
No, I wasn't thinking about Google Print
just then. I was thinking about something
called a "library." It's funny how authors
aren't complaining about those so much.
Of course, there is a difference -- libraries
pay for the copy they receive, whereas
Google will be paying nothing (to the
publishers, at least). But then again,
and this is the
point most journalists seem
to wallow in missing, no one can read a copyrighted book
directly from Google Print.
It's just a portal to find books;
not to read them. Buying or borrowing the book found
is what follows in the life of the happy user.
So, let's start from libraries, and modify it,
from the author's point of view, to Google Print.
We lose a tiny fraction from the publisher's
revenue (and hence an even tinier fraction of the
author's income), and we gain a gigantic leap in
How much is that publicity worth? Well, let's put
it this way -- how many web sites call Google and
ask them to not be searchable from their main page?
A huge number of web pages can be viewed in their
entirety from Google's cache - even pages whose
only source of revenue is ads - and I haven't heard
of any lawsuit from A Group of Concerned Web Publishers.
This is a call to abjure the nescient avarice of
the pusillanimous journalist. An automatic
card catalog which doesn't give away your content
(yes, that's what Google Print is), is a
good thing, for searchers and searchees alike.
||The Linear Algebra of Social Identity|
||This entry is inspired by the abundance
of inanity (=inaneness) inundating
* almost every
personal description profile to be found on
I propose an experiment in machine learning:
your corpus is the set of all profiles of
undergraduates on facebook. Step 1) establish
a list of common cliches, "I just want to be
myself," "live in the moment,"
"live each day
as if it's your last." (etc.)
Step 2) summarize each person by the cliches
in their profile. Step 3) run a clustering
algorithm to find which cliches go together.
Try to minimize the number of clusters without
sacrificing too much predictability of which
cliches are most likely to be found together.
(Added complexity: try to predict which cliches
are likely to be found in the profile within one
The result: a cluster basis of cliches for
the social identity world of the current
American undergrad generation.
My hypothesis: the basis is actually rather
feeble, bland, and depressingly uniform in
essence. The grand irony is that so much
attempted originality is just thinly veiled
emulation of the perceived creativity of
our pithless cultural avatars.
||I thought about updating this blog a
lot more often in order to make it
more interesting. Didn't work out.
(Compare this blog with the one
just before it, immediately below.)
There's just something about mass
which I find unappealing.
Novelty seems to me akin to superficial
beauty -- it's skin-deep, at best.
How many daily-updated blogs are
really interesting? I think very few.
Granted, even this blog is not that
interesting, but I prefer this style
to the more common, write whatever I'm
thinking whenever I think it near the keyboard
I don't quite understand the way
blog works (eg who really writes the entries),
but it illustrates my point.
So what else can a good blog have besides
novelty? Interesting content, obviously.
I think people also tend to look for
personality and a sense of confiding of
secrets in blogs. Perhaps by reading
someone else's blog, you feel more connected
to them. You gain a sense of true, contemporary,
and real companionship in a non-judgmental
way. If you really wanted to talk to the
people whose blog you read, you can do so
My goals in keeping a blog are probably a
little different than most other people.
Instead of expounding upon a single
theme such as politics or some private
industry sector or listing the details
of my day-to-day life or even talking about
my relationships, I want to express my
(relatively) abstract ideas. I would like
to record, in a sort of personality
driven manner, my pseudo-random cerebral
journeys down a diversity of curiosity-borne
||Novelty vs. content|
||So I did a mini-experiment to
try and see how often people read this
blog. The answer: not very often. But
then again, I don't add to it very often,
do I? So why should I expect the
interest to be high?
I say things on my site that I would
consider interesting to read (for
example, some ideas about abstract
But that (good content)
in itself is not quite enough
for popularity (unless you have
a huge amount of it, perhaps). It
seems that continuous novelty is
vital to a webpage which interests
What's so good about new stuff?
Old stuff is still worthwhile,
sometimes better for the time-testing,
and there's more than enough of it
to consume a lifetime.
Here is one hypothesis: consider our
`information-awareness' as another sense,
akin to hearing or vision. It is not
exactly a physical perception, but it is
still a type of perception -- of a slightly
more abstract nature. Humans desire
to be perceiving all that is going on
around them. We don't want to miss anything
that could be important. There is an
innate desire to keep our senses as
observant as possible. Reading about new
information is just another way of
satisfying this desire.
The interesting thing about this hypothesis
is that we don't care so much about the
particular information we're gathering.
When I read an online news article, yes
I care about the story, but in 95% of the
cases, I wouldn't be reading the article
if I thought it wasn't about something
very temporally fresh.
Is novelty over-rated? What do you think?
||The Insane King Puzzler|
A friend of mine, Anthony Hodsdon, gave me this
puzzle a couple days ago.
An insane king sits alone on an infinite chessboard.
He's probably crazy due to the vast overwhelming
loneliness of being the only chess piece on an infinite
board, much as Euler probably felt while exploring
the far reaches of mathematics he could hardly hope to
explicate to the far-inferior second-best savants
of the world.
In any case, the point is that our king is bound in
a straightjacket. And which way must one travel in
said garment? Why straightly, of course (being a king
he moves stately as well). This means the king cannot
move diagonally. Rather, he can move one square per
turn either up, down, left, or right.
Enter the advisary. A demoness removes squares
from the board one at a time. Once a square is removed,
it may not be replaced, and the king may not move
there. The king and demoness take turns, the king
going first (albeit a
The demoness wins if she can completely surround the
king -- that is, if she can remove enough squares so
that the king has finitely many places to move to.
Implicitly, the king "wins" if he has a strategy which
will indefinitely avoid such confinement.
The question is: who will win? Assume that both players
have infinite mental capacity, id est, that they both
play their own optimal strategy.
Of course, I wouldn't have been so eager to post this
if I did not have my own
solution ready as well.
||If you are insane|
||I was recently watching an episode of the old
British television series
The Prisoner in which the protagonist's mind is placed into
someone else's body. This led me to wonder how I would react to
such a thing happening to me. (By the way, I don't watch TV as a
rule, but I'm allowed to watch old shows if they're of decent
quality and commercial-free.)
So what would you do if you woke up one day to find yourself
in an entirely different body? Freak out is probably the
correct answer. How about something a tad more subtle, and
in a way more realistic: what if nothing seemed to have
changed to you, yet everyone around you kept telling you
that you were a different person? For instance, suppose that
you grew up as "Sue," and then one day everyone starts to call
you "Arnold." Certainly this would be more than a morsel
unsettling. I believe most people would simply insist
incessantly that they were who they believed to be -- id est,
Unfortunately, that would not be a wise course of action. If
everyone calls you Arnold, you must accept the role. How can
you get along in "the world" when you no longer live there?
You must learn to live in this new place where you are now
Arny (for short).
An even more interesting question arises -- what is the truth?
Who are you now? We have two subjective but usually
reliable measurements: what you think of yourself, and what the
world thinks. To be sure, this is a question worth a good
deal of philosophical ponderance.
I would say that you are in effect both Sue and Arnold.
You are Sue in that you know of Sue, and you also know that,
in your mind, you are the person who satisfies the essential
properties of being Sue. You are Arnold in that everyone else
sees in you some essential property of being Arnold; otherwise
how could they be so presumptuous as to correct you about your
Which brings us to the question of essential properties. What
makes something that which it is? What if two authorities on
identity disagree? I would say that essential properties are
in fact totally subjective. I will be bold enough to suggest that
all identification can occur only within a mind, subjectively.
Hence all essential properties are nothing more than psychological
guidelines which vary between minds yet tend to agree most of
the time due to the fact that we all live in the same world.
They are psychological guidelines which we have developed over
time to help us distinguish this one object from others similar
to it. (Subplot: we need comparison before we can enact
identificaiton.) And if two authorities disagree about identity,
it should be no great surprise as identity is no longer an
inherent property of the object in question!
Identity not an inherent property? Insanity indeed --
from two identities to zero (in five paragraphs no less).
||The When Game|
||This is a weird little game you
can play in your head when you're in
a new place on your own.
Suppose you have just arrived
from the future in a time machine.
The thing is, you built the time machine
last night when you were a little drunk,
and so it works, but it always sends you
to a random place and time (but still on Earth and
sometime in human history,
let's say). So each time you use your
time machine you have to figure out when
The trick is to forget your entire real
life and become absorbed with this time travel
daydream. Once you do that, you can look around
you with a fresh perspective, and observe all
the little clues which give away which era it
is, which century, which year, etc.
Looking at calendars or newspapers is basically
cheating, so don't do that. But consider things
like architectural styles, or the fashions of clothes
people are wearing, or the technology being used
nearby. If you can hear any music or conversation,
listen for clues about the current topics of popular
It all sounds very silly because, of course, you
know exactly when you are. But once you get
the hang of truly forgetting, the game becomes more fun.
It's really interesting to see which places are a dead
giveaway (e.g. busy, successful places with lots of people),
and which have no clues at all (such as a very old house
or building with no other people in it).
||It's easy to believe in the future
as long as it stays there.
||The Psychology of Wasting Time
||How exactly does one waste time?
The key, I think, is that it depends
on the perspective of the person you
ask. For instance, Alice might
consider it very worth-while to
play video games all day, while Bob
might regard this as an utter waste.
I wanted to answer this question so
I could make sure I don't waste my
So here's a theory: each person has
several contexts in which they see
themselves as existing. These form
identities for the person. For
instance, I see myself as a part
of the company I work for (this
summer), and a part of the
mathematics research community.
These two contexts give me a sense
of who I am, of identity. Based on
this framework, I think one feels
that their time is being spent
well when what they're doing is
contributing to one of these identities.
It's probably true that some of
these identities are more important
than others. It's also probably the
case that some contexts are harder to
contribute to than others. For
instance, I could easily contribute
to the community of my friends by
sending out an email. But this is not
quite as satisfying as publishing a
new math theorem. However, the math
theorem is a little harder to do. There's probably
even a correlation between difficulty and
satisfaction (although I can think of
one or two exceptions).
Yesterday a friend of mine (Steve Magill) gave me
this puzzle. I haven't completely solved it yet.
(Addendum: I solved it. I'll put up the
solution in a little bit.)
Thirteen pirates have collected a treasure they'd
like to keep in storage for a while. They are going
to lock it in a chest with several locks on it.
Each lock can be opened only by the right key, of
which there may be a few copies. Each pirate
owns several keys.
The pirates would like it so that no six pirates,
using their combined keys, could possibly open
the chest. However, they would also like any
seven pirates to be able to open the chest.
The question is: what is the
minimum number of locks necessary to solve
Just in case the description seems a little
vague, here is a sample solution for a smaller
case: suppose there are only three pirates. We
would like it so that no one pirate, acting alone,
could open the chest; but any two together could.
Then the chest need only have three locks.
Call these locks 1, 2, and 3. Give pirate A
keys to 1 and 2; give pirate B keys to 1 and 3;
and pirate C keys to 2 and 3. This works.
Think about why it's impossible to do with only