Re: A Problem To Solve :-

On 5/16/24 21:54, Physfitfreak wrote:

Physfit noticed the nice pullover again and got a bit curious. He took out his notebook and began jotting down some numbers and stuff. In about 5 minutes he knew exactly in how many different ways he was able to pay for the 19 Dinar pullover a few hours back.

Physfit was still within the two-hour period of lunch, therefore he took advantage of such rare moments in his life to ask for some coffee. He was served a cup of excellent Arabian coffee the likes of which the Americans had never tasted in their entire limited, slaving lives. Simply out of their reach. Unknown to them.
Then, you know what happened, don't you. That coffee thing kicked in bad ... And since a few other Houthi scientists as well had had some of that same coffee with him, they all got into a frenzy of excited talking about their projects and the successes and the problems they were still experiencing. One of such issues got Physfit's attention cause he knew something about.
Houthis had developed a missile capable of delivering and detonating nukes of fission type. For reasons of efficiency in production and the technology that they already had at hand and developed so far, the uranium core pieces, after coming together shortly before detonation, could form two different types of cylinders; one with a square cross-section, and the other with hexagonal cross-section. Other choices had not yet gotten developed.
The total mass of the two forms were equal. But Houthis were looking for ways to have _both_ options before those pieces would come together, based on what the target was. For some target areas they wanted the pieces to form a square cylinder, and for some other types of target areas, a regular hexagonal cylinder. This would make them use the power of the bomb more judicially and effectively.
On the other hand, the matter of precision in the various sizes in those pieces was absolutely critical, otherwise the bombs would not detonate optimally and in a totally predictable way. Therefore a very high degree of accuracy in cutting such pieces was required.
They had developed the machinery to do just that, but they needed to give the software running it, the exact form of sizes and angles involved, not just approximations of them. In other words, it was the computer running the machinery that decided the optimum accuracy, and it needed to be fed the exact form of what the length and angles were.
For instance, it wouldn't work right, if you'd give it 1.414213562, or even 1.414213562373095048802 to mean square root of 2. You would have to give the computer sqrt(2), and computer then would decide to what level of accuracy to interpret and use the number. This was so because tiny inaccuracies in sizes of lengths or angles had to be compensated for by judicial use of the ac-curacies elsewhere.
In other words, Physfit understood that their issue boiled down to the problem of how to divide up a square into minimum number of pieces, in a way that same pieces would then be either reassembled into the original square, or, they could get reassembled into a hexagon of the exact same surface area. This was the issue that those Houthis were heatedly discussing and got Physfit's attention.
Based on previous experience in sci.physics usenet forum, where he had proposed and solved that problem a few years earlier, Physfit knew this problem had a nice solution that would provide the exact sizes and angles of various cuts on the square, to yield a regular hexagon of identical area after same pieces were arranged differently.
Physfit:  "This problem of yours has a solution."
Houthis suddenly went silent and turned to him.
Houthis:  "Are you sure of that? Remember that we need exact expressions of lengths and angles of the dissected pieces involved, not just measurements made on existing samples using micrometers or protractors. We all know how the square should be dissected into minimal number of pieces to yield a hexagon with them, but this issue is about knowing the exact forms of such lengths and angles, and not an approximation of them."
Physfit:  "I understand, and this problem has a solution because I solved it in 2020 a couple of months after Covid lock-down began."
Houthis:  "Great!... All we need is those exact dimensions and specs! Everything else is already done. How soon can you provide them to us? Say, the square has a side of d. We need the dimensions and angles of the dissections exactly expressed as functions of d only. "
Physfit:  "Well, as you must've experienced it yourselves, it is not a trivial problem. In 2020 it took me about a week to solve and derive the exact dimensions, starting from zero. I would think it will take, now, a couple of days, if not sooner, to re-derive those numbers, cause I have at least some vague memories about what strategies and methods took me to the solutions."
Houthis:  "A couple of days is great! May we ask you to proceed and do it?"
Physfit:  "Absolutely. Is there a motel of some sort around here? I need to first get a few hours of real sleep. A nap on a chair won't do it."
Houthis:  "We are under British and American daily bombardments. We would not recommend using a motel, or staying outside of this compound for more than a couple of hours. This compound itself has very safe underground rooms and offices for our scientists and engineers to spend their time in while busy with their projects. You're quite welcome to use one."
Physfit:  "Deal! I'm going to need some stationery items."
Houthis:  "No computer?"
Physfit:  "Not really. In case needed, I do have my own Aspire One mini laptop dedicated to DOS with qBASIC on it, in my backpack."
He then added,
Physfit:  "I also want a flask of that same coffee we just had, but with no cardamom added. I'll be outside my eating period and do not wish to trigger my insulin production system."
Houthis:  "How long will you go without eating?..."
Physfit:  "If nothing unusual happens to me, about 48 hours. I should have your specs in term of d by then."
Houthis:  "Great. We'll see you when your next eating period begins then. What would you want us prepare for your meal at that time?"
Physfit:  "Same food I had in that restaurant 2 hours back. A full course of Mandi with two bowls of Fahsa on the side, please :) "
Houthis:  "Excellent choices. Consider it done. Will see you in 48 hours or so."
Physfit was led to a few levels lower down in that compound and into a corridor along which various doors had the Arabic names of various scientists on them, and a few Persian names and one Chinese name among them as well."
The place resembled a typical American research center where you'd read familiar names of scientists on the doors as you walked along such corridors, but in this case, in that corridor, none of those names were familiar to Physfit. He'd been away from such activities for decades.
Two days later, he stepped out of his room, fresh, showered, rested, and clean, and hungry for that Mandi and Fahsa like he couldn't even dare to think of it, discovering that someone had placed a name on his door reading, "Dr. Physfit Freak."
"CRAP!", he said out loud. "This sucker cro-magnon title doesn't want to leave my fucking life to myself."
"NOT EVEN In YEMEN!..."
"Not even in some quirky bizarre dream. Fuck you cro-magnons :-( "
"Fuck you for that!"
In his hand he had a few pages of drawings and calculations, the last page of it giving the exact derived dimensions of the dissected parts in terms of a general d, the square's side.
Show the work for deriving those exact forms of dimensions and angles, each in terms of a function of d only; d being the side of the square.
(not trivial)