Triangle With Rectangle Insert: CR4 Challenge (08/25/09)

Oh! You're supposed to take the peel off those suckers

Compass? Straight Egde? I think i have seen those things in a museum just joking

This is really off topic, but i have a cartoon that has a little kid laying on the ground coloring and his older brother looking at him a jealous like and saying "Wow....real crayons on paper! I can only draw with a computer program."

Unfortuantly i think a lot of skill was lost when we stopped teaching how to draw something on paper........oh well such is the way of technology

Compass? Straight Egde? I think i have seen those things in a museum

How true. We put a man on the moon with a compass, square, and slide rule. I have every expectation there are few people under the age of 60 who are able to operate these ancient tools of calculation.

i will admit that i learned how to use the compass and the straight edge.....even a t-square....i learned to design on the board before i ever learned to design on the computer.....my first design job was on a board.....and i am only 43 but the slide rule, while i own one, i never learned to use it....it was a gift from my uncle, calculators are so much easier......

Thank you. =TeeSquare=

I liked this sufficiently to follow jim in trying to vote you back on topic.

That said, I deal with three of your issues by using a spreadsheet (either Calc or EXCEL). That leaves the original entry open to inspection and makes repeat calculations extremely easy; it allows you to display the answers to the appropriate number of significant digits.

BTW, in the days of slide rule and tables I needed two rules: a straight rule for quick calculations where confidence to 3SF was good enough, and a large cylindrical rule for up to 5SF.

I'm possibly being dim here, but I don't follow your comment on safety factors meaning that calculations don't need to be precise. The 'most obvious' counter-example being the length of a strut used as a brace; however, in a competitive situation on a large project even a fraction of a percent over what is required to meet safety fators can significantly affect profits.

Maybe I'll find time to think about the challenge question...

Maybe I'll find time to think about the challenge question...

I've been thinking about it, but it hasn't got me anywhere _(yet) !

<Teesquares comments are very worth a read, and I've voted him up too.>

Thanks for your thoughtful comments.

I must admit to being stubbornly old-fashioned! Ready access to a computer is not automatic for me. I started using one primarily as a glorified typewriter, and hardly got beyond using Word97 with Win98 on an assembled contraption. Now I am struggling with Linux/Firefox/OpenOffice on a laptop. Keeping up with current software, hardware, inter-compatibility and anti-virus requirements is not an affordable option for me. I just muddle through somehow with a minimum of stuff on board, because asking for help might solve an immediate problem, but with strange effects elsewhere. No one can explain how or why things happen in a computer -- seems like mostly trial and error to me. I avoid adding new software for fear of unexpected consequences.

I've seldom used Excel or Calc, though I know the benefits of quickly comparing a lot of alternative options. I just don't have the patience to reformulate my problem (equations, headings, intermediate results ...) in a 'computer-friendly' manner. If I need to do it regularly I guess I'll learn or be forced to. I've been doing manual 'spreadsheets' all along -- perhaps before Lotus was created -- and it gives me an insight into how the variables behave when I write down each step. Not so obvious when Excel or Calc displays everything instantly. Obviously I'm not efficient or productive, but those who are can easily miss important details. Making things easy has its downside. I just prefer to be wary of computers, and actually dislike staring at a screen (I don't possess a TV set).

I don't know what a cylindrical (slide?) rule is -- will search google when I find time. I suppose 3SF and 5SF are statistical terms for levels of accuracy/confidence (a subject I've never grasped).

I'll try to explain my comment on safety factors. Taking your example of a strut, the formula for column buckling presumes a perfect geometric shape and exactly known mechanical properties for the material. In practice there will be dimensional variations, initial bend or twist, eccentricity of loading, uncertain end conditions, etc. Different authors and codes of practice do not have a uniform view on how these should be accounted for. It is also common for items outside the tolerance limits to be used without getting detected. In a 'large' project with thousands of identical struts under identical load conditions, it may be prudent or cost-effective to test two or three pieces to failure before finalising the design. But my experience has really been limited to one-off machinery design, where economics and competitiveness were not the primary criteria. Sometimes wild imagination was necessary with just a hope that it would work reliably. There are generally no benchmarks for comparison.

A formal calculation may have to be submitted to meet a legal requirement. That does not necessarily make a design safe! Or unsafe. For instance a gear pair which is known to operate quite satisfactorily may very well 'fail' a calculation test according to AGMA or DIN standards or both. There are major uncertainties with regard to material properties and operating conditions. It is impossible for a design code to provide fine-tuned factors to yield precisely safe results under all conceivable situations. Some tables of factors involve choices or judgements which can make a lot of difference -- like deciding between severe and very severe duty. A designer can blindly follow a code and wash his hands off further responsibility. If he has the option (as I frequently had) he can apply his mind to the problem in a constructive way (which can possibly lead to delays and disputes). Pardon me if I've overstated my point of view. To put it briefly, I believe that insight or intuition is necessary when using a text book or code of practice or computer programme to solve an engineering problem. And it still remains an uncertain solution. I guess very few will agree.

The larger issue is the extent to which technology (more so medical technology) is being hijacked or controlled by the legal profession and insurance industry. Fortunately, in my country we are still disorganised enough for competent persons (and, sadly, charlatans as well) to to take independent decisions without much fear of legal consequences. My doctor does little more than use a stethoscope, look at my tongue/eyes, poke my ribs/stomach, and check my blood test values before declaring me fit every few years. If I went to one of these new 'corporate' hospitals for a 'general checkup', I'll be subjected to a battery of costly procedures, and the doctor who diagnoses the reports will surely discover a lot of ailments whose treatment will put me out on the street. But globalisation is making rapid progress here too, and I'm hoping to kick the bucket sooner rather than later!

Hope I haven't exhausted your patience =TeeSquare=

You should read passingtongreen's comment (and vote to take it off "off-topic" and give it a GA). Assuming that "the chosen length for the hypotenuse" in the challenge means chosen by the solver and not someone trying to make the problem harder, this is the best solution so far. It is easy and exact.

Thanks,

Jim

Yes, I totally agree. I have voted Passingtongreen's post #72 as not off topic and I will give a GA if I get to do so.

The original problem states "The hypotenuse length being any combination of sizes taken from the rectangle"

I initially, like most people, interpreted this to mean that the hypotenuse chosen by the solver must be a length AB = L*CE + M*CF + N*CK where L, M and N can be zero or positive integers. However, JDRetired did not explicitly state this.

Now, after reading Passingtongreen's post #72, I would say that by dividing down the longer side by factors of 2, he is legitimately taking a valid " combination of sizes from the rectangle".

In post #18 jdretired writes "The hypotenuse length was to be a practical length that can be determined with out using a measuring instrument, FE is the horizontal distance or width of the rectangle, CF is the vertical or height of the rectangle, CE is the diagonal of the rectangle, or any other dimension that can be determined from the rectangle is legitimate".

I bring attention to the words "or any other dimension that can be determined from the rectangle is legitimate".

I look forward to your reponse jd

Hi MPM.

Yes I agree with your analysis, the spirit of the whole thing was to construct a triangle who's hypotenuse touches the corner of a rectangle, and the rectangle and hypotenuse sizes being of your own choice, the remark concerning the use of the rectangle as a reference was intended to be in keeping with not using a measuring device, how one arrived at the length AB was completely up to the individual, the end result being to have a hypotenuse who's length was known before starting the construct. The combinations shown where only there to get the ball rolling.

Regards JD.

"The rectangle and hypotenuse sizes being of your own choice.."

That is a much easier problem than the one most of use were attempting. If we construct our own rectangle, passingtongreen's proposal together with jim's extension does the job perfectly (constructing the rectangle for ourselves allows us to ensure AB satisfies the length constraint without making any estimates or measurements).

I agree.

This is why I believe Passingtongreen's answer was the best answer given and deserves a GA.

You said in response to Jim how do we know the hypotenuse gets longer if we increase n - measurement?

Well, we already know how to construct the perpendicular to CE - surely if we increase n as per passingtongreen's method until the hypotenuse is visibly greater than this perpendicular, then we have an exact solution.

Or am I missing something else here?

I think it's a matter of taste. I would assume that if greater-than and less-than measurements were forbidden, the equivalent estimates should also be forbidden. The ability to construct your rectangle and choose your length avoids this issue.

One reason I did not believe this to be the intention is that the calculation at the tail of the challenge bacomes irrelevant.

You can indeed define lengths for which there is a neat and exact construct. Your have defined a set of such lengths. Unfortunately, it does not appear to be one of the lengths that was defined in the original challenge.

N.B. that as the challenge does not appear to define which of the possible diagonals AB should be, I think you could with justification swap CF and CK if the relative sizes were in the "wrong" order for your simpler proposal.

That was me - I didn't realise that I was not logged in. I also forgot to welcome you to the Old Stager's club. Welcome

For what it is worth, my view is that you should not mark a post off-topic if there is something significant that is on-topic. If you feel that there is too much that is off-topic to leave it unmarked, it's probably best to separate the on-topic bit into a separate posting.

If only we were all as meticulous as that all the time! But the upside is that CR4 is very forgiving - especially if at least some of your contributions have value - as in your case.

I don't like to rain on someone else's parade. So, if I've got a side issue like this one, and don't think it worth a new thread, I post off-topic. I think this is interesting, but I don't know how many others would.

Agreed - it is well established (and accepted) to leave the separate posting in the thread that stimulated it

Thank you PVP45 and Physicist for a very interesting exchange, the points you have raised I have taken on board, and hope should I post any future question? I get it right, but though I tried very hard this time, apparently I did not. It was once pointed out to me the purpose of these questions is to enter into the spirit of what the author intended. A positive or negative contribution is a valid comment.

Regards JD.

I was reading the diagram quite literally, with the long direction of the rectangle, horizontal, and the angle above the diagonal of the rectangle to be greater than a right angle. If we, instead, say that the angle must not be ninety degrees, the simpler solution holds in either direction. Thank you for your welcome, in your following post.

Gratifying to learn there are still folk around who can appreciate the old methods. In my case it was mostly constrained access to modern gadgets in the circumstances of my working life which dictated my dependence on simple tools and slow procedures. Maybe I just never grew out of that mindset.

My aversion to computers is because I see so much of misuse and dependence on them, which I suspect is commercially driven. Competitive pressures apparently make it impossible for today's organisations to provide technical staff (engineers, designers, draughtsmen ...) with the kind of workshop or field training , which was routine at one time.

Coping with developments in computers and internet is a constant nightmare for me -- especially now as I try to work with Linux and OpenOffice (instead of Win98+Word97). I just about manage a quick look at CR4 every other day or so and save the page(s) if possible for viewing later.

It was dumb of me not to figure out that physicist?'s 3SF and 5SF referred to significant figures (I invariably face difficulties with abbreviations and acronyms). I had no idea what a cylindrical slide rule was until I looked up wikipedia after seeing it mentioned here. (Haven't figured out the mechanical construction yet.) I can see that he needed the greater accuracy for scientific and mathematical calculations.

The point I had tried to emphasise about engineering calculations was that in most practical problems (mechanics, strength of materials, fluid flow, heat transfer, ...) the available data or assumptions may involve an uncertainty of anything from 1% to 10% if not more. As you mentioned, a lot of choices are already standardised in steps (bolt sizes, gear modules, pipe diameters, motor ratings ...) and one just chooses the next size. At the end of a calculation we should of course trace back and check that all the choices were mutually compatible. In this process any more digits than a slide rule can provide represents but imaginary precision. I am inclined to endorse the following quote from the wikipedia article on 'slide rule':

"The spatial, manual operation of slide rules cultivates in the user an intuition for numerical relationships and scale that people who've used only digital calculators often lack. Since you must explicitly note the order of magnitude at each step in order to interpret your results, you are less likely to make wildly wrong errors. You are forced to use common sense and an understanding of your subject as you calculate. Since order of magnitude gets the greatest prominence when using a slide rule, and precision is limited only to the few digits that are normally useful, users are less likely to make errors of false precision."

Concepts like safety factor or margin of ignorance are mostly used in a subjective way by engineers depending on their experience and intuition, familiarity with similar problems, risk perception, and so on. Text book examples and codes of practice may serve as guidelines (or be mandatory) in somewhat routine situations. In an open-ended problem it is unlikely that two persons will follow the same thought process. I think the problem with computer software is that even a bad solution can be presented in an attractive and convincing format with pretty coloured pictures. The bugs are hidden away somewhere. My grouse with some 3D modelling software I've encountered is the absence of dimensions -- I'm only comfortable with proper views and sections in which all the details are apparent. Guess I'm just ranting as usual.

It is quite embarrassing to see just my lone post#8 appearing as an almost GA, when there have been so many expert on-topic contributions which tax my understanding -- and I've not quite understood the question yet! I agree with MathsPhysicsManiac #38 about this being a stimulating challenge (strictly observer status for me, besides off-topic comments).

=TeeSquare=

I did know what the cylindrical slide rule was, but if we needed more accuracy we used "Smoley's", (Smoley's Four Combined Tables For Engineers, Architects and Students (Logarithms and Squares, Slopes and Rises, Logarithmic Trigonometric Tables, Segmental Functions)).

I did spend some time to see if there was some iterative way, along the lines of "Hardy Cross' Moment Distribution".

I think the lack of a fixed answer has made for more enjoyment.

Back to the problem.

Agree completely with you... the ABSOLUTE value of a quantity is seldom an ABSOULTE necessity. So what if I calculate the volume of concrete required as 8,396.156 ft³, the mix plant sells it to me by the cubic yard!

There is a discussion going on Forum Thread Problems with Pi that bangs around this idea. There is, actually, some hair pulling going on.

Lots of off-topic posts on this challenge question... maybe I better have another look, and see if I can contribute something useful.

Many thanks, JD. I shall have a play with this over the weekend. Even if I'm unable to come up with a decent solution proposal, I shall enjoy reading the thread as it develops.

I see you come back from your August bank holiday just a little bit sweeter.

Regards JD.

Not a relevant comment (either way). I interpreted the avoidance of elucidation as obscurantism, and I did not anticipate it from your quarter. Grrr.

Any and all dimension are included, as long as the vertical length and horizontal length and diagonal length can be determined and referenced, (required for the maths) and the rectangle can be dissected to find any other size you require. Hope you enjoyed your break.

Regards JD.

Oops! Much embarrassed by all the superfluous publicity over a patently off-topic post. Perhaps I asked for it. But I was just responding to a couple of remarks which struck a chord in me.

I should apologise to jdretired for further diverting attention away from the challenge question.

The funny thing is that my few GAs were awarded mainly for off-topic limericks and such, in the 'fixing the pump' thread. For me the certificate from jim35848 and comments by physicist?, Kris and Doorman are worth more than a crop of GAs. At least I'm satisfied that there are people on this(?) planet who do not treat ALL my outlandish notions as complete rubbish. There's just the niggling doubt about where CR4 is located. (Some asylum in cyberspace?)

Now this is clearly off-topic. =TeeSquare=

Apologises not necessary, I was one who voted that you where not off topic. I enjoy physicist involvement and Kris and doorman and jim35848 who first responded and yourself, I'm an outside the box thinker and and roll with positive and negative responses, just keep them coming.

Regards JD.

The "fly in the ointment" is that if it stays open on its own, it's also a divider, which is a measuring instrument. In traditional Euclidean geometry, dividers are a no-no.

You are of course theoretically correct. But I was under the impression that the reason modern geometers have resorted to a fixable compass it that you can use a Euclidean compass to do anything that a fixable compass will do - it's just the operation is a lot more complex. (Dividers will only measure out integers - just about the easiest thing you can do with a true Euclidean compass and a straight edge)

Not exactly. A Euclidean compass can only transfer a line by swinging the arc. You can indeed limit your fixable compass to that task and you can do things like the familiar construction of a perpendicular. But, one of the things you absolutely cannot do is use the compass to mark (or compare) a distance alongside a straightedge. That creates a measuring instrument. And, you can use dividers for much more than integers.

All that is apparently missing from the collapsing compass is the ability to create a circle centred on an arbitrary point of length equal L to a predrawn line XY

But in fact that can be done as follows:

You state that you know how to draw a perpendicular to a line that passes through a defined point. Use two such perpendiculars to creates OS parallel to XY and another two to XT parallel to OX. The intersection Z of these two lines is a distance L (=XY) from O

Set compass and draw circle.

Using this method, every time that you wish to use a collapsing compass to transfer a length you need to draw a parallelogram. It's tedious, but it does the same job as a fixable compass.

No, I understand that and easily concede that is OK to do. But, you cannot hold a collapsing compass alongside a straightedge. It may seem a pedantic, picky point, but it makes all the difference in solving equations.

"There's a workaround, but it limits construction to transferring a line and prohibits comparing line lengths."

Clearly, the construction I described will allow us to say which of two lines is longer - which is what I would mean by "comparing line lengths". So your meaning must be different; can you help with some detail?

I agree that the collapsible compass cannot be used as a sliding measure - but I don't believe that either dividers or a fixed compass will do that anyway.

Don't parse too much. I've seen you show your great familiarity with Euclid. I think I remember you even having a book of the postulates.

You may, of course, check lines for an inequality. You may not compare lines in any sense that equates to measuring.

You write: "You may not compare lines in any sense that equates to measuring."
I fear that I would regard that statement as misleading in this context. It is of course true that Euclid was only concerned with relationships that were exact, which means that any approximation in measurement is outside his terms of reference. But you what you previously wrote was that the tools (collapsing compass and unmarked straight-edge) are incapable of performing what we would regard as measurement. This is demonstrably false.
All that you need do to measure the relative lengths of two lines is to construct a fixed "ruler" based on repeating an integral fraction of the length of one of the lines, and then transfer the length of the other line onto the ruler with one end on one of the markings. You can in principle achieve unlimited accuracy/resolution by repeatedly multiplying any residue.

And, again, I believe you are familiar enough with Euclid to know that you cannot in any sense create rulers. It is thought by some that other geometers permitted rulers, but Euclid did not (or so we think). Some geometers will go so far as to insist that you may only hold one instrument at any one time - either the compass or the straightedge, never both.

I agree that once you allow "marking off", there is no limit to accuracy/resolution (in principle). That indeed is the foundation of any measurement system. And, once I can mark off distances, there doesn't seem to be any limit to my computational possibilities; certainly, I can then build an analog computer. Even a transcendental equation becomes simple to solve.

For example, one of the two exact solutions I know for trisecting an angle requires that I hold the compass in one hand while turning the straightedge with the other. It is exact. It does violate Euclid.

Why do I care? I think geometry one of the important bodies of knowledge and one that is rapidly vanishing. I don't suppose I can do anything about that, but I will, in my bastardization of Thomas, "Rage, rage, against the dying of the light.".

Cheers.

I think geometry one of the important bodies of knowledge and one that is rapidly vanishing

How I would love to give you a GA for that ! If somehow it were possible to allow students the time to play with Euclidean construction, their knowledge would be so much better. They might find themselves 're-inventing the wheel', or pursuing impossible goals, but play-with-it-and-see is surely the best learning method. Reading endless tomes on theory has it's value, but time spent with blank paper and and abstract problem is never wasted.

You're more than likely correct in your assumption of what Fyz knows of Euclidean construction, but is not the greater point here learning ? Personally I feel that JD's initial description might have been a little clearer, but it really doesn't matter. JD has clarified, and it's up to individuals to draw (sorry for the pun) what learning experience they can. I've not come up with anything I consider worth contributing to the question, but I'm still enjoying it. The value of questions such as this one are not in the finding of a solution, but in the process of thinking.

Debating the finer points of Euclidean construction is quite valid and topical, but it's not the be-all and end-all. Much thanks to you all for an interesting read.

I agree about the larger point. That's why I'm posting "off-topic".

Based on this discussion, it would also be good if they were to experience the difference between Euclidean construction methods and the more narrowly defined (provable) Euclidean geometry.

To misquote: "yes you can" (create rulers). Abandon your preconceptions and look at the evidence - better still, try it using Euclidean construction techniques (realising that this is not the same as Euclidean geometry - see below) and tell me if-and-where you find a difficulty.

I do agree that trisecting an angle in the way you mention uses the tools in a way that Euclid would have regarded as unsound. But (to labour the point) measurement in the manner I described can be done while using the tools in precisely the way that Euclid would have envisaged.

Of course you cannot move the marked ruler about in any convenient wholesale fashion, or combine its use with the compass - as this latter would destroy the setting of your (fragile) compass. But you can readily create a marked line ("ruler") that is any rational multiple of a pre-existing length. The most convenient way to use this would be at a fixed position on your papyrus, and then copy the length-under-consideration to your ruler.

The reason that this would not appear in Euclid's work is that the work only concerns relationships that are "provable" as exact. Anything involving measurement of any kind is by definition approximate.
N.B that I can't see how your computer would be analogue - the best I could manage using measurement is equivalent to a successive approximation digital method.

An impasse perhaps? Since the answer has been provided, perhaps another place, another time?

perhaps another place...

please, please , please.... don't openly discusss where with any degree of exactitude !

I think he means the House of Lords.

Fortunately, I'm not qualified in that respect, so I'll try once more to see if TVP is willing to resolve it here - so at least neither of us will ever mislead anyone on this topic again.

That's a very charitable definition - perhaps Parliament should move with the times and adopt the phrase "...another planet". I suspect that if you and TVP continue much longer, we'll shortly be hearing the phrase, " I refer the honorable gentleman..."

Yes, this is one occasion on when I'm quite happy to assert that neither of you are qualified in that respect - dissection of a Challenge Question is a much more tricky game than our glorious leaders could play ! I'm not privy to knowledge on JD's tonsorial endowment, but if he feels like I do right now he'll be checking the coefficient of friction up top. That is ,of course, presuming he's not brushing brick-dust from the forehead ! I jest too much - an interesting bit of debate between all. The question, and JD's solution are going to take me some time to get to grips with. Not that I can use the excuse of other tasks (though it's true enough), there are many subtle points at issue to ponder. Personally, I think that JD has posed a very good problem for us. It's not easy to come up with a concise proposal that answers it _{(in my case no proposal at all !).}Putting forward a question is often a much greater art than solving one.

Can you first clarify your position?

Are you saying that measurement against a scale is not possible using Euclidean construction techniques? or
Are you saying that Euclid explicitly disallows the use of this sort of measurement?

If it's the former, basic integrity would suggest that you either check where the technique I propose breaks down or admit that you aren't as certain as appears.
If it's the latter, I have looked (again) at a direct translation of Euclid's Elements, and can't find anything to that effect*. That said, the content of the Elements indicates that accurate but inexact measurements are irrelevant to the intent of the work - i.e. proof by example is not valid in this context, and only exact constructions are of interest.
*It is possible I've missed something you know about - if so, please give a reference. (Book#, Definition# or Proposition#, etc)

I don't agree. Euclid used a compass and a straight edge to construct a perpendicular bisector of a line segment. This requires making arcs of equal radius on both ends of the line. He also constructed a parallel line through a point which required setting the compass distance from the intersection of an arc and two lines. None of the construction techniques used here to solve the challenge question require anymore than 1) drawing straight lines through the intersection of lines and 2) drawing arcs of "measured" lengths centered at the intersection of lines.

I think TVP has agreed that we can transfer lengths using parallelograms and circles.

But to create the parallelogram requires arcs of equal radius without a transfer method - fortunately, we can create perpendiculars through a point by extending the line and marking an arc of a circle to cut the line, and then we can use the distance between the arcs to set the radius of the compass used for the perpendicular bisector.

I managed to start on your first method - as I understand it, this too starts with a successive approximation; so it's even more painful than the second - and still not an exact construction.

I didn't challenge any of the solutions. I pointed out what I think is a legitimate obstacle to solving the question. I am slowly working my way through some of the solutions, but am not challenging them.

No. I think I have determined that none of the solutions satisfy the challenge. Perhaps I misunderstood the challenge, but I believe the conditions to be:

AB must be a combination of the sides and diagonals of the rectangle.

AB must be longer than the perpendicular of CE.

No measuring instrument may be used.

Are those incorrect?

If that is what you were saying, I agree:
The challenge as posed has not been solved. In fact, it cannot be solved, as it asks for a line of any of those specific defined lengths passing through E - so measurement instruments would not help.

Rewording the challenge to demand a solution that give AB to a precision of 0.001 units, and does not require confirmation, I also agree that no solution has been proposed solutions is sufficient - more about that later. If we were to rephrase "no measuring instrument" as "no measurement may be made other than those that use the compass and the unmarked straight edge as separate stages", we would still require the extra step of dividing the side by 10000 and drawing a circle to show that the error was less than this (it is a mater of semantics whether a single larger/smaller should classify as a measurement?).

To return to the question of a solution to the re-posed (soluble) "accurate to 0.001 without confirmation": the successive approximation approach (first given in post #19) can be the basis for this. As implied there, the additional piece of information that would be needed is the number of iterations required. That would have to cover the worst case, so it either requires an analytic solution (very tough) or (more likely) a spread-sheet demonstration of the worst region (very tedious).

But perhaps we could now say that passingtongreen has shown a route to a solution to the challenge as posed - exploit the ambiguity of the presentation and choose the length so that it is one that you can fit. (Unfortunately, his proposal does not necessarily result in a length that is longer than the orthogonal to CE).

It would have been nice if I was right, but alas its not the case, the challenge as proposed has not been solved, I did not state how wide the rectangle was, or high it was, or how long the hypotenuse was, how can there be an answer? such abstract questions, one has to be able to apply one knowledge, not come up with logical conclusions? of inferred facts, such depth of perception has to be admired.

Regards JD.

I realised that the puzzle does not have a fixed answer, so I read it as a request for an iterative method that moved closer and closer to an exact answer, hence my saying that I thought I had "walked around" your real question. I was a little disappointed when I found, or thought I found, an exact solution.

It may have been more interesting because there is no single solution.

I too found that for certain rectangles, "his proposal does not necessarily result in a length that is longer than the orthogonal to CE". However, if we increase the the value of n to be larger, we can still get an exact solution. Each time we increase n, we make the slope of L steeper and therefore longer.

Thanks,

Jim

Agreed - but how do we determine whether the length is longer? Measurement?

You write:-

"AB must be a combination of the sides and diagonals of the rectangle."

but the problem wording actually states:-

"The hypotenuse length being any combination of sizes taken from the rectangle"

Yes, I know, but what other size could there be?

Hi TVP45

The concept of the challenge was to solve how to calculate the angle CAB, and test that the angle was correct to a given tolerance, the rectangle sizes and hypotenuse length being arbitrary, the challenge was about a construct that aided that endeavour, it was not intended to be a geometric proof, and the rules that go along with it, though that was unavoidable.

I would be interested to see if you can give me a calculated solution without the aid of the above constructs. A step by step mathematical solution? If you can I'll owe you a Guinness.

Regards JD.

I'm game. Can you briefly run over the challenge again? Several posters implied I didn't fully understand the original.

LOL- know where you're going with the 'vitruvian man', and iterative sequence always hold a fascination for me. I'll leave it to others to take that approach. Quite sure it leads to an answer (), but it's a case of ever decreasing circles. It's a good example of where to proceed on this question. Bitter experience has taught me the possibilities of disapeaering up yer own **** ! Yeah, have I been there !! A good point to make, and very valid in regard to 'solving' this querstion.

The 'ladder against wall' question I mentioned is out there somewhere. I'll check it later.

Oops, sorry, that was me !

The ladder is in the memory jog above, post #8.

Regards JD.

LOL - I really should check my own checking at times ! Cheers for pointing it out_{...simpers off muttering 'doh'}