Puzzles
Lance
By EDWARD ROTHSTEIN
Published: NYT April 3, 2006
ATLANTA - Two hundred green marbles are in a green jar, and 200 red
marbles are in a red jar. Thirty marbles are removed from the green jar
and put into the red jar, which is then shaken and stirred. Thirty
marbles are then scooped from that mixture and put back into the green
jar. Which jar has more of the wrong color marbles?
Simple, right? The scoop removes only a very small proportion of green
marbles, so more green remain in the red jar.
Except that's wrong: both jars begin and end with 200 marbles, so any
green marble missing from the green jar had to have been replaced by a
red one, and vice versa. The two jars end up with exactly the same
number of wrongly colored marbles.
That's an old puzzle. I had heard it posed about glasses of water and
wine, with a teaspoon of wine being added to the water, and another
teaspoon of the mixture spooned back. But even well-known puzzles
retain their power, as was made clear again and again last month at the
seventh "Gathering for Gardner." These conferences of mathematicians,
puzzlers, game-players and magicians at the Ritz-Carlton here began as
personal tributes to Martin Gardner, Scientific American's legendary
Mathematical Games columnist, and now take place without the master's
presence (he is 91). During four days of talks and tricks, the oldest
puzzles mixed freely with the newest.
Puzzles are a strong lure. So when the mathematician Solomon W. Golomb
discussed the marble problem as one of his "favorite quickies," or when
the mathematician Peter Winkler posed puzzles about people with blue
and red dots on their foreheads, or about prisoners doomed to die if
they don't find their names inside boxes, or when the Google software
engineer and puzzle master Wei-Hwa Huang explained how he should have
solved the puzzle that cost him the championship in a sudoku
competition in Italy last month, one willingly put aside concerns of
daily life.
It hardly mattered that there is no such thing as a tribe of truth
tellers, or a man with a pet monkey and a pile of coconuts, or any
other of those strange inhabitants of puzzle universes. There are no
sudoku number-puzzle grids in nature, either. We imagine them, along
with marbles in jars, or magic squares, or eggs covered with flexible
tiles. We stage esoteric treasure hunts and construct three-dimensional
models of four-dimensional objects. The only requirement is that
everything be clearly defined with a limited number of laws governing
behavior.
Puzzles can seem magical because they really are from a made-up
universe that is bounded and simplified. Within that universe, though,
free play is allowed.
That is why these conferences, organized with almost theatrical dash by
the mathematician Elwyn Berlekamp, the puzzle enthusiast (and generous
sponsor) Tom Rodgers and the magician Mark Setteducati also include
demonstrations of sleight of hand, feats of memory and exotic juggling.
They are also attracting more and more invited attendees (over 270 this
time, about 100 more than the last conference).
The mixture becomes surprisingly provocative. When the Swedish
physician turned card virtuoso Lennart Green seems to clumsily drop a
shuffled deck of cards and then shows how it magically organized itself
by suit from high to low, or causes any card a viewer names to fly out
from the center of a deck, order is created out of chaos, logic out of
confusion. Sleight of hand or sleight of mind? It hardly matters.
When Robert J. Lang, a laser physicist, talks with passion about
origami - the Japanese art of paper folding - the line between play
and discovery also completely dissolves. Mr. Lang is what might be
called an origamist (see www.langorigami.com). He wrote the book
"Origami Design Secrets: Mathematical Methods for an Ancient Art" (AK
Peters, 2003) and has created tarantulas, delicate herons, 12-spined
shells and big-horned elk out of single, uncut, folded sheets of paper.
This is maximal effect with maximal constraints and minimal materials.
In that respect, origami is like the puzzle about marbles or a riddle
about truth tellers or a game like Go involving black and white stones
on a square board: artifice with the ability to amaze. How much can you
understand or create when there is so little provided, so many
restrictions, and so much possibility?
Mr. Lang points out that while in 1950 there only about 100 standard
origami designs, in recent years a "mathematics of origami," studying
the theory of folds and constructions, has evolved. As a result, he
says, more than 30,000 origami designs now exist, and in September a
fourth international conference on origami will be held at the
California Institute of Technology.
Once restricted to domestic decoration, origami has also become useful
for designing everything from foldable tourist maps to expandable heart
stents. Mr. Lang has worked on the folding of air bags in a car, and on
a design for a collapsible telescope-mirror more than 300 feet wide
that might unfold in space.
This is also a legacy that Mr. Gardner leaves to generations of
researchers, teachers and entertainers: don't try to understand the
whole world at once. Take only a small part of it. Or better yet:
invent your own universe in which there are very few elements and very
few rules - a game, a puzzle, a theory. These circumscribed and
artificial worlds are like sheets of paper subject to the rules of
folding, yet they can yield remarkable results having almost uncanny
power. The science fiction writer Arthur C. Clarke once said that any
sufficiently advanced technology is indistinguishable from magic, but
maybe it's also true that any sufficiently powerful "magic" eventually
evolves into essential technology.
That magic is produced when one begins to see a baffling puzzle from a
different perspective: what once seemed impenetrable suddenly becomes
transparent. The effect resembles the images involving optical
illusions or Escher-like transformations of foreground and background
that are often displayed at these gatherings. The eye and mind look at
one thing and start to see another. Instead of seeing fish in the sea,
as in one famous Escher image, one gradually begins to see birds in the
sky. Instead of thinking about how many green marbles are in the scoop,
one thinks about the unchanging numbers of marbles in the jars. Solving
puzzles often means seeing double - an experience singularly magical.
And afterwards, one returns to daily life - absurdly confident that
some day it too will begin to make sense.
Dave Smith
in a red jar. Thirty marbles are removed from the green jar and put
into the red jar, which is then shaken and stirred. Thirty marbles are
then scooped from that mixture and put back into the green jar. Which
jar has more of the wrong color marbles?
Simple, right? The scoop removes only a very small proportion of green
marbles, so more green remain in the red jar.
Except that's wrong: both jars begin and end with 200 marbles, so any
green marble missing from the green jar had to have been replaced by a
red one, and vice versa. The two jars end up with exactly the same
number of wrongly colored marbles."
Could a computer be programmed to jump to the wrong conclusion as
people tend to do?
Lance
"Could a computer be programmed to jump to the wrong conclusion as
people tend to do? "
------------
Why not? Work out the "heuristic" that people use and then programme it
inb the computer... Simulation is a standard method for testing claims
about how people perform tasks.
Lance
Lance
"Could a computer be programmed to jump to the wrong conclusion as
people tend to do? "
------------
Peter.H....@gmail.com
inb the computer... Simulation is a standard method for testing claims
about how people perform tasks.
--------------------------------------
I once wrote a very successful simulation - it certainly got me out of
a hole and had people very happy with the results being so precisely
aligned to real life. Since I am fully aware of how it worked, I can
attest that it worked in a way that was utterly unlike what it was
simulating and relied almost entirely on fiddle factors (and luck) to
produce such good results.
Remember that, given any sequence of numbers (the output of your
simulation) you can produce infinitely many functions that will produce
them. That means that it is almost completely impossible for the
function you produce (that is your simulation) to be identical to the
one of your target system (1/infinity being rather close to 0). That's
just functions, you can, of course, also have infinitely many relations
that needn't be functions.
Of course, with real systems you don't have infinity, so there are just
a shit-load of alternative simulations that will produce the same
results - having the same results, in other words, has no predictive
value in deciding if the production of them was the same.
If you have other things that are similar, besides just your
simulation, then you have more of a chance. If, for example, you use a
group of a few thousand human beings (who all have brains quite
different physically, but fairly similar in operation) to simulate the
general human response to something you are likely to have quite a good
result quite often because your theory includes the notion that people
respond quite similarly to a lot of things. You'll be aware of how
often a few thousand people don't give a good result.
Lance
"Of course, with real systems you don't have infinity, so there are
just
a shit-load of alternative simulations that will produce the same
results - having the same results, in other words, has no predictive
value in deciding if the production of them was the same. "
-----------------
So who says you only write simulations? Most theories are tested
repeatedly, and using "triangualtion" - different classes of evidence,
evidence from different fields, or different kinds of studies - and
when the results converge we think we may be on the right track.
Dave's question was about: "Could you programme a computer..." and the
answer is "Yes".
Such programs can also be useful for research purposes in many
different ways.
Lance
Peter.H....@gmail.com
repeatedly, and using "triangualtion" - different classes of evidence,
evidence from different fields, or different kinds of studies - and
when the results converge we think we may be on the right track.
Another metaphor to the rescue! Triangulation is a mapping technique
that's useful on the surface of a very large sphere (relative to the
size of the triangulator). It is hopeless for use during earthquakes
and not much good at mapping the surface of marbles since they don't
have many church towers or other landmarks to work from.
The metaphorical use of 'triangulation' suggests that the different
methods work like different positions in a village, connected by their
views of the church tower. The metaphor conveniently ignores the fact
that many things (the operation of the brain being just one) are not
equivalent or even similar to villages on a planet, they are more like
villages during earthquakes or marbles - they don't keep still and the
metric you use for one direction doesn't necessarily give the same
result in another - triangulation using a crew with half metre rods and
half rod or perch rods and no calculators is going to take quite a bit
longer triangulating somewhere and are rather more likely to get it
wrong.
What I'm pointing out is that results may appear to converge but, in
fact, do no such thing. Log-log graphs were invented for people who
wanted to pretend that things that weren't similar really are, they can
be used to wicked intent, but it is worth realising that studies that
appear to converge might similarly be illusions because of scale,
metric, substrate or even simply because things are different at
different times.
Again, I'm not saying we should give up. I'm not saying that it is
hopeless. However, I am saying that being sure that computing is a good
analogy for thinking is far more likely to be wrong than right.
Cosmology is relatively simple compared to understanding the brain.
Part of the difficulty that cosmologists suffer from is just this
illusiory convergence. A number of things appear to be pointing to an
inevitable conclusion, but then it turns out that they're all secretly
(or not so secretly) based on the same set of assumptions and the
convergence is more a reflection of this than what is going on out
there. The discussions about Black Holes and Dark Matter in the New
Scientist are mainly entertaining for the manifold different ways in
which this constant characteristic of cosmology reveals itself.
Actually the two studies aren't quite that dissimilar. Both the human
brain and the stars are quite difficult to get to when in their
standard operating configuration (stardust is another matter, of
course, as Hair! Hair! pointed out so very long ago). That means that
you have to base a lot of faith on your instruments, your assumptions
and your calculations. Which means that, as sure as eggs came before
chickens, you're going to find illusiory convergence more than once.
Lance
Popper pointed out - can NEVER be sure we are right. So we do we out
best, we try different methods, we simplify, we look for convergences,
we repeat earlier studies, we read the critiques that other provide.
Maybe at the end we think we have advanced knowledge a little.
I couldn't actually give a stuff what your opinion of software
simulations is - I have seen some that I thought were useful, and I
hope more such will be developed.
Lance
Peter H.M. Brooks
at all like the thing that they're modelling though. Thinking something
useful and claiming that it is an accurate representation are quite
different matters.
Lance
"That's nice. It doesn't help the question of whether they are anything
at all like the thing that they're modelling though. Thinking something
useful and claiming that it is an accurate representation are quite
different matters."
-------------
And you can decide such questions on the basis of your little piece of
software?
Why don't you cite some real research and show us why you doubt it? Why
don't you offer some support for your position based on what is really
out there?
Lance
Peter H.M. Brooks
> Peter Brooks wrote:
>
> "That's nice. It doesn't help the question of whether they are anything
>
> at all like the thing that they're modelling though. Thinking something
>
> useful and claiming that it is an accurate representation are quite
> different matters."
>
> -------------
>
> And you can decide such questions on the basis of your little piece of
> software?
>
the point - they were pedagogic devices. That is decided by mathematical
, physical and philosophical facts about reality.
>
> Why don't you cite some real research and show us why you doubt it? Why
> don't you offer some support for your position based on what is really
> out there?
>
understanding and perceiving what is 'really out there' - that is
essentially what I'm trying to make clear to you!
A model and the thing that it models are quite different - just as a map
is quite different from the territory that it maps unless it is a 1-1
map, and even then it is different from the territory itself (it can be
folded, for example).
Even if you were to model yourself by creating a biological clone, then,
as you know, that clone would not only be different from you, but would
also behave and know things that are different from what you know and
behave - even if it was a Science Fiction clone the same age as you,
made by copying you at an atomic level - as it would have different
experiences from the moment it had a separate identity.
Models are certainly useful and the better they work the more practical
utility they have. It is a major mistake, though, to believe that the
model is the same as the thing it models. You can make a really good
engineering model of a bridge that may well give a good indication of
how the bridge will respond to stress and strain - but until you build
the real bridge, on real ground, in real air and let it survive in a
real environment, you can't know how it will actually behave. Certainly,
even with a fairly crude model, you can make predictions about its
behaviour that are quite likely to be close to its actual behaviour -
but, since you aren't modelling the real bridge you only have, at best,
a loose connection. That's even if you make a physical scale model -
even if you make an exactly to scale model in a different place.
The model is never the reality.
Once you move from a physical exactly to scale model of something (which
can't be an exact model) to something much more tenuous, a computational
or mathematical model, then your level of certainty that the reality
will match the model reduces even further.
As I say, this isn't a matter of research findings - though you could
find plenty that supply further evidence that this is true.
Lance
a person enabling homeostasis to take place. But I do think some forms
of balancing do take place in the body, and that these are real and not
models.
Similarly I do think the brain can and does compute and that even
though it is constructed from neurons and the like, one can correctly
describe the activities of the brain as computation. You yourself
pointed that this is trivially true when, for example, you perform
mental arithmetic. In what sense is this then a model?
Perhaps you think I am making the claim that the brain is nothing BUT a
computer - and the answer, Of course not. But the brain does compute,
and when it does, it functions according to the same limits as any
other computer - and does not transcend those limits by accessing
Quantum Mechanical processes, nor any other extra- material processes.
Cheers
Lance
Peter H.M. Brooks
> Well I am sure I don't think there is a chemical balance sitting inside
> a person enabling homeostasis to take place. But I do think some forms
> of balancing do take place in the body, and that these are real and not
> models.
>
give to the pattern we recognise in reality.
>
> Similarly I do think the brain can and does compute and that even
> though it is constructed from neurons and the like, one can correctly
> describe the activities of the brain as computation. You yourself
> pointed that this is trivially true when, for example, you perform
> mental arithmetic. In what sense is this then a model?
>
simply be a memory job, not actually computing through an arithmetical
algorithm. Of course a digital computer could be programmed to do it
that way too.
Computing is a model in the sense I explained under the travelling
salesman problem. The real salesman has many variables to consider when
selecting his route. The computer problem is a vast simplification of
this and is only difficult when it considers an unlikely number of
cities in this simplified case. In that sense it is a very simple model
that hardly begins to deal with the difficult job that the brain does.
>
> Perhaps you think I am making the claim that the brain is nothing BUT a
> computer - and the answer, Of course not. But the brain does compute,
> and when it does, it functions according to the same limits as any
> other computer - and does not transcend those limits by accessing
> Quantum Mechanical processes, nor any other extra- material processes.
>
It transcends the limits of the computer by dealing with genuine
real-world problems - robots still have trouble climbing stairs because
it is too difficult computationally.