Project Lovecraft Problem 2

[danbaron]

Problem 2:

Find the integer values of A, B, C, D, E, and F, such that the following six equations
are simultaneously satisfied.

A^1 + B^2 + C^1 + D^2 + E^1 + F^2 = 1429090927300389160

A^2 + B^1 + C^1 + D^1 + E^2 + F^2 = 1217848777792070180

A^2 + B^1 + C^2 + D^1 + E^2 + F^1 = 0938746011122624520

A^2 + B^1 + C^2 + D^2 + E^1 + F^1 = 0859834257193611164

A^1 + B^1 + C^2 + D^1 + E^2 + F^2 = 1272071906073384380

A^2 + B^2 + C^1 + D^2 + E^1 + F^1 = 1095765032349629300

DTB.

[ErosOlmi]

This is really hard, isn't it? ;--)

What is the prize? :D

[Petr Schreiber]

Very interesting problems,

one curious question -> why they are called "Lovecraft", I guess there is no link to H.P.Lovecraft :)

[kryton9]

Where are these sort of problems used in the real world. Can someone point out what A, B... F would stand for? I would also like to see an explanation for Problem 1.
I just want to know if I should wrack my brain in trying to understand this, in case I will ever use it for a future problem in programming. Thanks.

[danbaron]

No time now, guys.

Tonight, more.

Yes, Petr, H.P. Lovecraft.

I thought about using Kafka, but decided to use an American.

Dan.

[Charles Pegge]

The raw substitution method still generates nasty polynomials but you can simplify these expressions by subtracting pairs from each other.

Example:

Expression1 A^1 + B^2 + C^1 + D^2 + E^1 + F^2 = 1429090927300389160

Expression2 A^2 + B^2 + C^1 + D^2 + E^1 + F^1 = 1095765032349629300

difference the two:

A^1-A^2+F^2-F^1= 1429090927300389160-1095765032349629300


then trying to cancel out the square terms by other combinations looks promising.

[danbaron]

Probably, it is hard by the brute force method, Eros.

For the first problem, it took your machine approximately 1600 seconds.

The total possible number of iterations was approximately, 1.8*10^10.

For this problem, the total number of iterations would be at least 2*10^48.

So, statistically, this problem should take approximately 10^38 times as long.

10^38 * 1600 seconds = 5*10^33 years (approximately).

The age of the universe is approximately 1.37*10^10 years.

So, according to statistics, by the brute force method, this problem on your
machine, should take approximately 3*10^23 times the current age of the
universe.

(On the other hand, it is possible that you could guess the values of A-F, on
your first try.)

----------------------------------------------------------------

1st prize = thinBasic Version 2.0
2nd prize = an unexpected visit from Mossad

----------------------------------------------------------------

Hi Kent.

Say, we have the two equations,

2*x + 3*y = 8

5*x + 7*y = 19.

Each is the equation of a line.

They are called linear because, x and y appear by themselves, except for the
constant terms which multiply them.

The two lines intersect at the point (1, 2). If you put x = 1 and y = 2 in
either equation, you will get the number to the right of the "=".

Since we only have x and y, we are working in two dimensions.

But, the same idea works for any number of dimensions.

All linear mathematical objects are similar. So, methods have been developed
which systematize working with systems of linear equations. In the simple case
above, these methods show how to find the point (1, 2).

The methods work in almost every case.

The subject of this study is called linear algebra.

But, equations like the next two, are nonlinear,

x*y + x^2*y^7 = 9

x^9 + y^5 = 11.

Both of these equations define a curve in the x-y plane.

Generally, nonlinear mathematical objects have nothing in common with each
other. There is no general method that will solve any system of nonlinear
equations.

So, for the above two nonlinear equations, it is more work to determine if and
where they intersect.

For the problem I made, there are six nonlinear equations, and six variables,
A-F.

Each of the six equations is a surface in 6 dimensional space. It turns out,
that in this case, all of the surfaces intersect at one point (a, b, c, d, e,
f). The problem is to find that point. If you do, and you put the values of a-f
in each equation, then the total on the left of the "=", will equal the number
to the right of the "=".

What good is all of this?

Linear algebra is used in many scientific areas.

Concerning nonlinear algebra, for pure mathematicians, it is good just because
it is an area of mathematics which is not fully understood.

Concerning practical applications, many physical laws are nonlinear.

Each nonlinear equation in a system can represent a particular physical law.

A group of related physical laws may have common parameters, such as
temperature, pressure, volume, etc. Each parameter becomes a variable, in a
system of equations expressing those laws. Then, the system is solved,
for instance, to determine the conditions in which all of the laws apply.

Basically, Problem 1 was solved by guessing. Different values of V-Z were put
into the 5 equations, until for each equation, the expression to the left of the
"=" was the same as the the number to the right of the "=". That event
occurring, indicated that the problem was solved. The method is called "brute
force". The larger the number of possible solutions is, the harder it is for the
brute force method to work. That is why the strongest encryption algorithms use
256 bit keys. There then become 2^256 different possible keys. And, no matter
how fast you go, it would take too long to try each one.

(Too tired now.)

Dan

[kryton9]

Thanks for the explanations Dan. You do a great job in explaining this stuff.

One last question,
Each of the six equations is a surface in 6 dimensional space. It turns out,
that in this case, all of the surfaces intersect at one point (a, b, c, d, e,
f)., to visualize this, can I just picture 6 curves in 3d Space that all intersect at one point?

I made a picture to show what I mean.

[Charles Pegge]

Yes Kent, all the expressions (we hope) intersect at one point, except these hypercurve expressions traverse 6D space which is almost impossible to visualise directly.

As programmers we are of course very familiar with multidimensional space as a logical construct. The fields in a database or the members in a variable type can be regarded as true dimensions. It is just that we have no built-in faculty for perceiving more than 3 dimensions in space. So we have to do all this strange stuff with algebra instead.

[danbaron]

Absolutely, Kent.

Your picture shows it as good as can be done in 2D space.

And, I agree with Charles.

It seems impossible to us that there could be a 4th spatial dimension, perpendicular to each of the 3 spatial dimensions we seem to inhabit. But like Charles said, it could be that we just are not constructed to be able to perceive it (or them, if there is also a 5th, 6th ..).

---------------------------------------

It seems that some programmers like to try to solve this type of puzzle.

I think one of the reasons is that problems like this have exact (integer) solutions.

So, the programmers know that if they program the method correctly, then the exact answer will appear on the screen.

Either an answer, is right or wrong, there is no gray area.

Problems which require floating point calculations, usually can only be approximately solved on the computer.

I think there is less satisfaction then, because accuracy is limited by the computer's architecture.

And then there can be disagreement about what constitutes a correct answer.

[kryton9]

So just to be clear Charles and Dan, my representation is still not a true, a good approximation in 2D but not accurate to what is happening.
Is this because:
1. Each curve is in 6 dimension or
2. Some are in 3 dimensions while others in 4-6 dimensions?

I always thought the 4th dimension was time, so is this also in there somewhere?

[Charles Pegge]

Hi Kent,

Each of the 6 equations represents a set of points. Any point with coordinate values of A B C D E & F which make an equation true is a member of that set.

A point which has coordinate values which make all 6 equations true is what we a looking for. In graphical terms the equations are the multidimensional equivalent of lines.

To give a 3d example:

This equation expresses all the points that make the surface of a sphere with a radius of 1

x^2+y^2+z^2=1

Any point whose x y z coordinates make this equation true belongs somewhere on the the sphere's surface.

To further restrict the set of points we can add another equation restricting them to lie on an xy plane:

z=0

Taken together these equations confine the points to a circle.

Another equation will further restrict the points:

y=0.5

leaving only 2 possible points:

z=0, y=0.5, x=0.866
z=0, y=0.5, x=-0.866


With regard to time, it can be argued that it is not a proper dimension since you can only translate forwards (and occasionally backwards!) or scale them along the t axis but You cannot for instance rotate objects in the x t plane.

[kryton9]

Thanks Charles for a great explanation. I see now that each curve since it has A-F is 6 dimensional.

Perhaps for others confused like me, instead of a,b,c,d,e,f
substitute.

Points that make a curve:
x, y, z, a, b, c
Dimensions:
1, 2, 3, 4, 5, 6

[danbaron]

No time, now, maybe more later.

But, along with what Charles said, I think,

x^2 + y^2 = 1, indicates a 2D sphere with radius 1 (a circle with radius 1),

x^2 + y^2 + z^2 = 1, indicates a 3D sphere with radius 1 (what we normally think of as a sphere with radius 1),

x^2 + y^2 + z^2 + a^2 = 1, indicates a 4D sphere with radius 1 (for the mutually perpendicular axes, x, y, z, a),

x^2 + y^2 + z^2 + a^2 + b^2 + c^2 + d^2 = 1, indicates at 7D sphere with radius 1 (for the mutually perpendicular axes, x, y, z, a, b, c, d).

And, if the number of mutually perpendicular dimensions is infinity, then the sphere would be infinite dimensional with radius 1.

[kryton9]

That is cool, thanks Dan. A good addition to what Charles wrote about.

[danbaron]

And, if you think about it, a 3D sphere of radius r, contains an infinity of
circles of radius r. Each circle has three points in common with any other
circle, the center, and two points on the boundary.

Likewise, I think, a 4D sphere of radius r, contains an infinity of 3D spheres
of radius r. Each 3D sphere has the center point and one boundary circle of
radius r, in common with any of the other 3D spheres.

We are three dimensional, and live in a universe with three spatial dimensions,
as far as we know. But, we can imagine, that we instead were two dimensional,
and lived in a plane, like on the surface of a sheet of paper. If we were two
dimensional, then we would be unable to imagine how there could be a third
spatial dimension, z, perpendicular to both the x and y dimensions of the plane
we inhabited. In that case, whenever we moved from one place to another, and we
encountered an obstacle, we would have to go around it. We couldn't go over it
or under it, because, there would be no up or down. Let's say, a three
dimensional object passed through our plane world. How would we interpret it?
Say, the object had the shape of a torus (doughnut), and passed through our
plane edgewise. What would we see? A tiny ellipse would appear in front of us.
It would grow, and then as the hole reached the plane, the ellipse would split
into two ellipses. The two ellipses would become two circles as the torus
reached halfway through the plane. Then, the sequence would reverse, as the
torus exited the plane. While there were two ellipses, it would be impossible
for us to believe that they were both part of the same object, connected in a
spatial dimension, that we did not have access to.

What would happen if we were able to move in the z direction, even the tiniest
bit? I think we would find ourselves on another plane. Everything from our
previous world would be gone. It would be like living on the surface of one
sheet in a stack of paper, and then moving to the surface of another sheet.
Only, the sheets would have zero thickness, so any movement at all in the z
direction, would put us on a different sheet.

How would it look to us if a four dimensional object passed through our three
dimensional space? In that case, I think a three dimensional object would
suddenly appear in front of us, growing from a point. As it grew, it might
continually change shape. It might suddenly seem to split into multiple objects.
Then, the sequence, would reverse, and it would disappear. If we saw multiple
objects, we would not believe that they were all part of the same object,
connected in a higher spatial dimension. It seems to me, that if a 4th spatial
dimension existed, and we moved in its direction even the slightest bit, we
would effectively find ourselves in a different universe, analogous to a
different plane in the previous example.

But, these are only my speculations - from, The Twilight Zone!

Dan

[kryton9]

While taking a nap this afternoon, this popped into my head.

Imagine we are the size of bacteria. We live in a 3 dimensional space. Now imagine other beings the size of Suns and not bacteria. They too would be 3 dimensional space. But to each other it would seem they were not in the same dimensions. So perhaps, Scale is something to consider when we run into these more than 3 dimensional space?

[danbaron]

I agree, our entire universe could be a speck of dust within another universe.

------------------------------------

How about these?

1.
The universe is expanding.

What is it expanding into?

Since it is expanding, does that mean it is contained in something else?

Since it is expanding, it must be finite, correct?

2.
What is the shape of the universe?

Since it is finite, it must have a shape, right?

3.
Some physicists say that the universe could have been spawned by a quantum fluctuation.

If nothing existed before the universe, then what could there have been a quantum fluctuation of?

[kryton9]

I hope there will always be unanswered questions. Luckily so far it seems, the more we learn the more questions we have, this is really great. I hope it never ends!

[zak]

Hi
i believe in the idea that there is a multiverses; every universe have poped up to existence from the (greatest sea: whatever it is) in a quantum event, like this picture:

http://img84.imageshack.us/img84/6/pwexp11107sk2.jpg

indeed i feel dizziness when looking to this picture exactly like looking down from a hight building. (the picture i think it is from scientific american magazine)
in scientific american there are many articles about multiverses and the quantum fluctuations. some of them are philosophical .

[Charles Pegge]

When a star burns out, it collapses under its own gravitational mass to form a very dense body which then becomes unstable and explodes releasing huge amounts of matter and energy.

The universe seems to have a fractal structure so we might say that an exploding star re-enacts the Big Bang, and suggest that Big Bang is part of a cyclical process, rather than a single event.

Charles

[kryton9]

I couldn't sleep thinking about this stuff. Wanted to get this out of my head so perhaps I can sleep then.
I will use 6 dimensional sphere as in recent posts, because it really helps to visualize things.
These keep popping into my thoughts, scale, time and position.

Scale well we have our scale, measured in inches and feet or centimeters and meters. We have the Galactic scale measured in 100,000+- lightyears. We have universal scale again measured
in billions of lightyears. Then we have molecular scale, quantum scale... you get the idea. In each scale it is hard to see where a point from that scale is on a point in our scale. If you model in 3D
applications, this becomes really apparent when you import objects from another app. Depending on the scale, if too big or too small you don't see the imported object although it is in the scene at your scale.

The same holds true for time. Everytime I stand someplace or wait, I always think who stood their yesterday, or an hour earlier. Who will be there tomorrow?

So the 6 dimensional circle can be in two forms:

1. ourScale:x, y, z and otherScale: x, y, z so that at the same Time, 2 spheres of immensely different sizes have one shared point on the circle in our scale. The OtherCircle is either too big or too small for us to make out another point on its surface.

2. ourTimeAndPosition:x, y, z and otherTimeAndPosition:x, y, z. This would be 2 circles of same size occupying the same position in different times. So ourCircle would be here at this location now, but the otherCircle would be at this position either in the past or future. Again we don't see it because of the time we are at the same position is different.

Now this is for just the six dimensions.

We could have xyz for scale, time and position of 2 or more objects to make an infinite dimensional nonlinear equations then.

Anyways this is the way an artistic gut feeling turned into a vision from my point of view. I don't know if it will ever jive with the math, but it is the best I can do from going crazy thinking about this stuff :) Now hopefully off to sleep, it is 6:52am. Yikes!

[danbaron]

My brain is worn down from thinking about this stuff.

I like that picture a lot, zak. To me, it qualifies as beautiful art.

I think that humanity will never learn everything in our lifetimes, Kent, and probably, never ever.

Some famous person said something like this, I don't remember who, I think he was a scientist:
"Not only is there more in the universe than we know, there is more in the universe than we can know.".

I agree with that statement. It seems to me that humans arrogantly assume they are capable of understanding everything, without the slightest evidence that is true. I think there may be problems that are so far above our capability, that we are not even capable of recognizing them. If we can't recognize a problem, then for sure, we can't solve it.

About what Charles said, some stars when they burn out undergo gravitational collapse, and become black holes (maybe we are talking about the same thing). The gravitational field is still there but all of the matter, I think has shrunk to a point. A black hole sucks in anything that crosses its event horizon. I think that some people have speculated that gravitational collapse in one universe, could produce a "big bang" of another new universe.

When I am at a place, I also think about how tomorrow, it will be the same place, but occupied by different things. I think that is how time can be considered to be the 4th dimension of our universe. To know about the state at a particular location, you need the 3 coordinates of its point, and you also need its time "coordinate". I also think of time, as the dimension in our universe which permits motion. Nothing can move in space, without moving in the positive direction of time. If time is frozen, then so is everything else.

[Lionheart008]

I like this topic. I have written some years ago short stories about scifi themes ;)
we are all 3d-people ;) we cannot understand 4th. dimension. "time" you cannot feel or hear or taste or see. we have no borned arganism for this one. living objects of 5th. dimension are perhaps energy livings, I don't know. they can understand how we little ants living / acting on a such beautiful blue planet earth and don't understand what's around us or "above/below" us and we destroy our planet. don't forget it's unique! we have still this planet and cannot create a copy.

if a star collapse, you will get a white star or black hole, dying stars are pulsing. lights and energy power (modified energy with lightyear speed) will get no chance to live in a black hole. may be there is an exit or outpuptchannel to another universe (like stephen hawking suggest), but I think every material will substract to a singular object and explode after a while (for example in thousand years for a new planet/star).

there is a interacting multiverse I am quite sure and it's expanding and/or decreasing. big masses of galaxies or even "small planets" have big amount of gravity, so I can assume they have a certain influence to each another. our "milkyway" is a galaxy of more than 400 billion (milliards) stars and more, andromeda is bigger than "milkyway" and some day it will collides with with andromeda nebula galaxy. what will happen? :) lifetime of our "sol" sun is limited. you can imagine what will happen when this great star will die.

our planet isn't round, it's like the shade of an unrounded potato. we live nearly 80-90 years age long at this place. we are programming a better world with ideas, some people doing with practical works good things to make easier and better life for all or using thinbasic. but to understand how god or another entity created universe or think about it like a automatic generated system it's just a philosophical thougth and be worth to think about it, but I cannot real understand how my neighbor is thinking about his life or family or his job or his intention to buy a new audi or ferrari. How I can understand my friends? How I can understand my dog or my wife ? the BIG questions are used in daily motion called real life. it's enough to handle for me :)

it's important for many people to know how does this life function. where we come from, how much time you have at this place and what's sense of life (the life itself!). it's good for me to see what's the macro cosmos (universe, galaxy, stars, energy, gravity, time, dimensions) and micro cosmos (nature, cells, water, air, earth, fire, insects, human beeing and so on). you can think a lot about these aggregate situation you're seeing yourself. you can try to bound it all into mathematical aspects and functions. one side is exactly defined, other side is exactly "chaotically" so that's my opinion. every part needs each other. you can debate more than one day about this topic ;)

My idea is that universe is "chaotically AND exactly" defined like our sol system with planets they're cruising around sun or big gravity point. there is an undefined "black material/energy" they are keeping and attracting all together (stephen hawking). nobody will change that. it's a equal general definition for all elements and energy systems.

interesting themes for exploring. but we haven't time enough to explore universe with space ships or transmitters to jump to another planets or places. it's a pity :) so I explore my body, my mind, my friends, my wife, my house, my surrounding area and try to do my best to give them "positive power" :D to change the things in a good way...

thanks for your ideas danbaron, charles, kent, zak, good night, frank

[danbaron]

I agree, Lionheart, no one knows how it feels to be anyone, except himself.

Everyone has a unique viewpoint which no one else can fully share.

I know I can't understand my dogs and cats. I just try to do what seems to make them happy.

And, in the age we live in, we usually don't have the luxury of worrying about the big questions.

Doing what we need to do in order to survive this particular day, keeps most of us quite busy.

Sometimes I ask, why am I me and not someone else?, but no one ever answers.

[kryton9]

Interesting how something based on a complex math problem could lead to philosophical and then metaphysical thoughts. I guess this is the math for this sort of stuff.

[danbaron]

Here is the solution for this problem.

Dan


' Empty CONSOLE script created on 02-18-2010 18:42:34 by (ThinAIR)
Uses "CONSOLE"
Uses "FILE"

Global A As Quad = 109483301
Global B As Quad = -549673081
Global C As Quad = -257312497
Global D As Quad = 884102903
Global E As Quad = 927658179
Global F As Quad = -587632953

Global C1, C2, C3, C4, C5, C6 As Quad
Global S1, S2, S3, S4, S5, S6 As String
Global FT As String = "0000000000000000000"
Global OUTFILE As DWord
Global FILESTRING As String

Function TBMAIN()

C1 = A^1 + B^2 + C^1 + D^2 + E^1 + F^2
C2 = A^2 + B^1 + C^1 + D^1 + E^2 + F^2
C3 = A^2 + B^1 + C^2 + D^1 + E^2 + F^1
C4 = A^2 + B^1 + C^2 + D^2 + E^1 + F^1
C5 = A^1 + B^1 + C^2 + D^1 + E^2 + F^2
C6 = A^2 + B^2 + C^1 + D^2 + E^1 + F^1

S1 = Format$(C1, FT)
S2 = Format$(C2, FT)
S3 = Format$(C3, FT)
S4 = Format$(C4, FT)
S5 = Format$(C5, FT)
S6 = Format$(C6, FT)

Console_WriteLine(S1)
Console_WriteLine(S2)
Console_WriteLine(S3)
Console_WriteLine(S4)
Console_WriteLine(S5)
Console_WriteLine(S6)

FILESTRING = "LOVECRAFT2.TXT"
OUTFILE = FILE_Open(FILESTRING, "OUTPUT")
FILE_LinePrint(OUTFILE, S1)
FILE_LinePrint(OUTFILE, S2)
FILE_LinePrint(OUTFILE, S3)
FILE_LinePrint(OUTFILE, S4)
FILE_LinePrint(OUTFILE, S5)
FILE_LinePrint(OUTFILE, S6)
FILE_Close(OUTFILE)

WaitKey
End Function