gnpatterson

No you dont use an exponential equation, you use the assumption that each generation is the same make up as the previous the equations you make are for the probabilities that certain things will happen the probability that some thing will happen is based on the population distribution and the population distribution is based on the probability that certain things will happen. then you use a self consistency arguement In some ways i like these type of problems in other ways i hate them they work on the basis that you can solve them if you assume you can solve them on the other hand it requires more confidence than I normally have.

DQW are you the teacher of these kids?

sorry I am effectively (with decades of neglect) approaching this subject from the point of view of a novice. and for other reasons i find it difficult to understand the rules of forum etiquette ( well the rules of etiquette period. )

There are two questions that need to be answered 1) in the condition that the liquid is significantly less dense than the gas-- Is there any difference in the physics of the "upside down liquid" for example if you bubble heavy gas through the light liquid and the bubbles fall to the bottom surface and they burst out of it, as they burst out does the spatter and pop of the bubble look the same a when we do it the normal way round. 2) in the condition that the liquid and gases have the same density Does the physics vary from that seen in microgravity. We have all see astonauts squirting water into "zero gravity". When viscousity and surface tension are matched however it should be possible to see if a shaken container of droplets even looks the same and to probe it precisely by looking at how it scatters light. The point is that while we can easily answer what should happen it would a circumstance in which our assumptions could be tested in extreme and the possiblity of new phenomena might arise. as a micro-gravity experiment it would be interesting.

looking at your results I think I can spot your mistake. you have accidentally equated the "?" with the second square on the top row. NB you should not trust the relative sizes of the squares on the diagram this means while the square on the top right hand corner is shown as being smaller than the one below, it is in fact bigger. To solve the problem you have to go all the way to equating the total width and total height AND to finding the smallest set of integers for all the squares, this includes the small ones at the bottom. Care should also be taken to make sure that none of these measures slip into the negative integers if you have rashly canceled them out. You can only get the value of "?" if you solve the whole thing unless you take a leap of faith. btw did you get y=3x yourself? If so how did you do it without working out the whole thing, it doesn't seem possible?

the debate is "pie in sky" vs "sour grapes" the trick is to know ahead of time if you are the fox or the crow. I really think it is only worth doing if you are going to get enough out of the effort to make it worthwhile in any case.

As already mentioned the point of the "problem" is that it is so easy. It is the opposite of the one about naming the colours what it says about how we read is interesting in the debate about how to teach people to read. do you remember all that time you were taught to "spell the word out"? learning phonetically? interestingly in the UK the government is considering imposing "blended phonics" as a reading method to all children. i loved the use use of the word "defrag" to describe what the brain does when it appears to unscamble the middle letters. What it does however is just skip over them. what is happening is that your sensory system has been trained to do a job using an optimal parrallel processing system while you consciousness models it all as a continous single naritive. what puzzles like these do is show how your own (instinctive) model of what the brain is doing is so badly wrong. another notable use of this exact phenomena is the use of the FCUK advertising/logo in the UK by the company "French Connection UK". A totally artificial name used get those initials. It is so funny to watch the people who are similtaniously so anal as to be offended and at the same time so pedantic that they buy into the "proper" meaning of the initials. They are torn between being offended and insisting that the "initials" be read correctly; it is a hoot.

why would you want to cheat? why would you want to be a member of a highQ society (at all)? if you are not smart enough to solve a problem your self why would you want to join a society that would just give you more problems that you could not solve yourself. I have to agree with atinymonkey, the puzzles are copyright and property of the site. Just giving the answer would spoil the puzzle for any one wanting to do it for real. HOWEVER Some of the puzzles are of dubious provenance. The 24 squares is a published discovery of a group of mathematians. The Doughnut problem also published. The standard of puzzle is hardly going to separate out the "highly gifted", they are merely a pasttime that some rather egotistical and stupid people are going to use to impress themselves with. Groucho Marx: "I don't want to belong to any club that will accept me as a member"

Ha Ha furbles.co.uk is of course a red herring If you want to research the question "population dynamics equations" should form a useful google search

Without answering the question I will discuss how to solve it. the main issue in these quesitions is to form a self consistency equation with the various population types across a number of generations if necessary. Then solve the equations. you need to form the equations across enough generations to include all the historicity of the system. The creatures in the question are two distinct populations but the problem I have is in parsing the information given and in getting to grips with exactly what the sequence of events is. there does not appear to be any restriction on the territory so the solution is one in which a stable proportion of the two populations are exploding exponentially The life span of the creatures seems to be infinite which helps get rid of complexity The issue that I am stuck on is deciding on the factors that influence the interaction of the two populations, I suppose i have to assume some sort of territorial lottery with all creatures getting an equal "A priori" chance (this excepts the displaced sharers) As far as the dynamic stability of the system is concerned it seems to me that the system is likely to be stable as sharers have their "get out of jail free card" and effectively expand "ahead" of the dominators. the spacial distibution of the individuals is of course not relevant.

oh yes the close pack layer spacing is simple pythagorus to get the height of a unit tetrahedron The height of the triangle is sqrt(3)/2 then construct the triangle with this as the hypot and 1/3 this as the base, the height is then the number you require. this would be so much easier with a diagram

Had a look at the square question, it is not really fair because I recognised it straight away, it is the disection of a "perfect square" see http://mathworld.wolfram.com/PerfectSquareDissection.html were it is one of the examples given I really can't imagine getting many of the questions "right" without taking a large amount of time. Of course some of them I also recognize as mathmatical problems that I can do easily but that is hardly a fair test for example the cannon ball one is just a bit of discreet mathematics. others would seem to just require a significant amount of work the decryption type ones. most just seem to be based on the difficulty of having to find the correct strategy to avoid the hard work. How long are you supposed to take for the test? the squares one would be possible to do without "knowing" the answer again beacuse it is a bit of discreet maths that has to set up very carefully for it to get the correct answer. the diagram gives you the template to use.

I have often wondered if it is possible to have a liguid that so light and a gas so dense that the liquid is bouyant in the gas and if so what phenomena (micro-gravity style) you might observe. Or would it be just a exotic lava lamp?

I think I see what you are saying but could I confirm that have got it right by asking if you really mean K<<1? Wouldnt it be more meaningful to say the 1/K >> 1 as K cant go negative and zero doesnt qualify as "very much less than" one?

how do i charge? use paypal??

thanks johny321 I visited the site and the "free IQ test" gave the usual flattering results, allowing me to join the site. HOWEVER i agree with grocho marx "i wouldn't want to be in a club that would have someone like me as a member" BUT what is the 24 square question? it doesnt seem to be in the puzzles section of the site, or arent i looking in the right place?

I am not yet convinced that 594 _is_ correct. Obviously I have nothing better to do ALL day, but most of the time was spent typing the conversation about how to think about doing it. also I did a few su doku puzzles in between. the 594 solution has a lot going for it in that the 12 layers use up 11* 8.1649 + 1 = 9.98146239 of the 10 inches available. This gives very little room for manouver and would appear to lock this solution in place. HOWEVER this is not enough to _prove_ that another configuration is not possible. the mathsworld article references a paper http://www.combinatorics.org/Volume_11/PDF/v11i1r33.pdf which shows the actual arrangements for some spheres in cubes but for very low sphere numbers <40 were some computational work has been done to get solutions. it is obviously not likely to be able to assemble a workable simulation of 600 odd spheres. how did you come up with the problem?

As already pointed out a machine that extracts energy from a difficult or obscure source is not a perpetual motion machine. The technical definition of the two types of machine being given before are useful but I would like to see the adoption of the pschological defintion. It is a machine, the imagined construction of which, brings a benifit that is sufficient to seduce a person into making efforts beyond the level of their ablity to accurately estimate the likley of their success. I'm not totally happy with this definition but I like that it can be used to include things like time machines, faster than light travel etc. In particular I am interested in the relationship between how we generate such machines and the use of narrative in the thinking process. It seems to me that the construction of such machines depends first on the mental construction of a narrative involving the principal/discovery. Out of our day dreams these machines seem to force their way. You are asking yourself "what if I could" and then you have constructed enough of the mental picture to dazel yourself into forgetting that the thing you started from was a day dream. It has got to be a fundamental feature of brains that have only just evolved to use tools that such machines keep being designed.

Now looking at the actual problem here are some results close packed layer spacing = 0.816496581 spacing of rows = 0.866025404 sphere packing results http://mathworld.wolfram.com/SpherePacking.html if spheres could be packed without consideration of the walls then a volume of 10*10*5 could hold 628.5393611 spheres = 500 * packing fraction / volume of sphere HOWEVER considering real packing schemes 1) 12 layers of 10,9,10,9,10 and 9,10,9,10,9 gives 570 2) 5 layers of 10,9,10,9,10,9,10,9,10,9,10 and 9,10,9,10,9,10,9,10,9,10,9 gives 523 3) 12 layers of 5,4,5,4,5,4,5,4,5,4,5 and 4,5,4,5,4,5,4,5,4,5,4 gives 594 considering the amount of apparent wasted space in these "close packed" schemes I have to say that a "random" packing arrangement MUST exist that can do better BUT I cant say that it could be found easily I would like to try some different orientations of the layers but don't see this is easy enough to do.

try http://www.everyscience.com/Chemistry/Inorganic/Ionic_Solids/a.1296.php

OOPs I seem to have my hcp and fcc totally mixed up, it is other way round!? although the terminology is wrong confused the geometry is probably OK can anyone with a better notion help (in my defense it was many many years ago that I did this stuff)

I denigh saying that hcp is the solution! the problem that I see with hcp is the layer positions If you think about the the hexagonal layers hcp repeats in the form ABCABCABC while fcc repeats ABABABAB what is he talking about? you may well ask. the best way to show this is is with a diagram, here goes nothing __A_____A_____A_____A _____C_____C_____C __B_____B_____B _____A_____A_____A __C_____C_____C_____C _____B_____B_____B __A_____A_____A_____A _____C_____C_____C now if you print that out and join all the A's up you should have a hexagonal pattern, same with the B's and the C when you pack hexagonal layers you can pack the layers in one of two equivalent positions. If you draw a close packed layer and look at the equilateral triangles formed by the holes one set points up (or left) and one set points down (or right). You can then place the next layer either on all the triangles pointing up or the triangles pointing down. NO BIG deal at this stage HOWEVER when you place the third layer you have a choice, you can place it either back in the same position as the first layer (ABA) which is in triangles pointing in the opposite direction OR you can go on to place in the triangles pointing in the same direction (ABC). This is easiest to see if you build a model This does not effect the packing fraction in the large scale It does however effect the packing in a cuboid Firstly choice of which direction to pack in is critical for the purposes of demonstration I will consider starting with a layer on the 5 by 5 face of a cuboid. In the first layer we can pack in rows of 5,4,5,4,5 using up a total height of 4.464 = 4* 0.866 +1 The second layer of 4,5,4,5,4 reaches to 4.75277675= 4.464101615 + 1/3 0.866025404 If you choose the third layer to be type C (hcp) it would be 5,4,5,4 and the next row would be absent as if present it would push the height to 5.041451884 HOWEVER the fcc option would give you 5,4,5,4,5 ie a total of 23+22+23 for three layers rather than 23+24+18 (hcp). This analysis does not calcualate the "thickness" of the layers nor does it say that this is the optimal approach for the cuboid give 10 x 10 x 5. However the appearence of 5.04 which is so frustratingly close to 5 is suggestive that the problem is going to be interesting. It would in reality be possible to add that fifth row (third hcp layer) if you allow the spheres to rise up above the layer slightly or raised the whole layer up and back a bit.

Ah an interesting question. brings the memories flooding back, packing fraction results, fcc, bcc and hcp. I remember the results for large volumes where that face centered cubic and hexagonal close packing the accepted packing methods HOWEVER I also remember that for limited volumes psuedo-randomn packing based on a frustrated icosihedron can be better for the cuboid you suggest body centered cubic might even be the best. I cant see any easier way to answer the problem without trying the options out with some marbles and a cardboard box. but I'll think about it first.

sorry for misleading you (not intentional) what I meant was that when I started to do the "forwards" research to solve the puzzle I started to learn about the existing number including the mcnugget numbers. I started with the Catalan numbers then I came across the the page of "Miscellaneous Special Numbers" http://mathworld.wolfram.com/topics/MiscellaneousSpecialNumbers.html hence the McNugget Numbers eventually I found the key to your sequence http://mathworld.wolfram.com/PlaneDivisionbyCircles.html I realalised straight away that the gap was due to the selection of prime numbers as the base of the sequence. It would been neater if you had made it that the number of circles was prime. It is thus really just coincidence/chance that I found the answer. The mcnugget numbers were a bonus of the journey I felt worth reporting

In mathematics there is a conventional form of proof, ie. proof by contradiction You can prove something is true by taking the opposite and showing that it leads to an impossiblity. The technical discussion of perpetual motion is really a pointless one since it really ends up by saying that if energy can be created from nothing (rather than from a unknown source) energy cannot be measured, if it is not conserved, then it simply does not exist as a quantity and is not real (like money). However the idea of perpetual motion is interesting more as a psychological phenomena. the hope in peoples minds that has been generated by the idea over hundreds of years is evidence of something that has to happening at a biological level. it must indicate that humans are evolved to keep on trying things they know wont work, it is like thumping a broken television when you know it isnt even plugged in. I know I have done things knowing they wont work but being unable to stiffle the hope, I found the action happening almost without willing it. It is so natural in humans that people have conned investors out of money when the people handing it over have said, "we know it wont work but if it did we will be very rich" Humans can similaneously doubt and believe something at the same time and invest their money or biological future in the possibility that they are wrong. they will deliberately invest in risk to the point of stupidity. They are not the only animals that do this. an experiment with turkeys found that they would open containers that they would be effectively punished for opening even when they had "learnt" not to do it. It just illustrates to me that the evolution of life depends on a great deal of conterintuitive processes. It may in fact be hopeless but what is even more amazing is the attempt by people to reason with the "believers". Again this seems to be an incrediblily difficult to understand action. It is so altruistic, to try and teach another person miles away through a bulletin board that there is a way they can improve their life (i.e. by not wasting it in activities that over-reach the likely success). These people are equally failling to learn the lesson of history and science that you cant stop people creating pertpetual motion machines. PS here is my arguement against perpetual motion machines (it is not serious) If you could create a machine that created energy you can also creat a machine that destroyed energy. Imagine a machine that if pushed energy into it, it would disappear, no heat generated, no sound spilling out just energy going in and being destroyed. (also no mass being created, i don't want you thinking this is a "real" black hole) What would happen to the universe if such a machine was created? Enough powerful enough of such machines could destroy the universe forever. So I beg you, if you do discover perpetual motion, keep it quiet, you could be on the verge of destroying us all.