Listener 4114 - Three Square by Elap

[Philoctetes]

No - I haven't quite finished it yet. Still, one of the joys of a Listener comes when you learn something new, and primitive triangles are new to me. I assume that the numbers in the threes are also the genuine "answers". I do love the mathematicals!

[thebuzz999]

For a primitve RIGHT triangle it is true that gcd(a,b,c) = 1 if and only if gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. This is an immediate consequence of the Pythagorean Theorem. However, this is definitely not the case for scalenes. So you cannot conclude that just because the sides are pairwise relatively prime in primitive right triangles that the same is true for scalene.

Another way to look at it is to consider all triangles with integer sides. These can be partitioned into separate sets by similarity. Each partition has a unique smallest member - those are the primitive triangles. Example: one such set is {(2,3,4), (4,6,8), (6,9,12), ..., (2n,3n,4n), ...} (2,3,4) is the primitive one even though gcd(2,4)=2.

[sohcahtoa]

I seem to be stuck! Could I please ask for instance is the answer to P2 11?

[Mysterons]

sohcahtoa - if P2 is 11 then E3 is 333, but 333 cannot be the middle term of a primitive pythagorean triple:

http://aleph0.clarku....oyce/numbers/pyth.pdf

[fyellin]

I must say I found the main part of this puzzle to be a bit of a slog. It took a little bit of insight to get he first three digits. After that, it was mostly just keeping track of what newly discovered digits went into which clue, with more Python-code looping and searching than deductive reasoning.

That said, I found the denouement of this puzzle astonishing. An idea came to me, I told myself "that's just not possible!", and then gasped in awe when I saw that Elap had, in fact, accomplished the impossible.

[JackDeCrow]

Hah - well, I look forward to having that idea! Only really got down to this yesterday after a busy weekend, and bumbled through to a completed grid eventually. So am now where many folk seem to have been Friday night / Saturday. Hey ho - still a week of potential staring at it to go. (And if it's any consolation to those a bit cross with Mysterons, I'd already tried both those things and am still none the wiser!)

[logophile]

Now I'm intrigued ... what is Python-code looping please fyellin ?

[Clamzy]

Just finished after some days of most enjoyable solving. Looking through the thread I see we have the honour of a conribution from a setter using his "real" name. We have had setters here in the past such as Angrysolver and Listenerite but not using their setter name. I shall strictly observe Elap's request and offer not the slightest hint.

I will however jump the gun on Andrew and offer a bit of feedback. I solved this without recourse to spreadsheets, excel and the like. All I used was the calculator on the Windows desktop for the longer calculations. However I believe that JEG doesn't have a computer and he is an all correct solver as well as Listener scrutineer. The Listener setters' guidelines imply that the level of knowledge needed for a number puzzle should not exceed that of the old O'levels. What does it say about the technology required? Had I not had a computer I certainly would not have finished yet.

[logophile]

The modern scientific calculators have a longer display Clamzy, so this one was possible without recourse to a computer, fortunately for JEG.

[jamesah]

Two days working out all the clues which was reasonably enjoyable but no idea about the last part. I don't quite understand why the setter is so adamant against any hints appearing- I think that it is more satisfying to solve after a hint or two than not to solve it at all. No PDMs if I never get to the solution! For those who submit their answers I am fairly sure that they would not look on here anyway until they have sent off their solution.

[cruciverbali]

Send me an e-mail if you need a hint jamesah:

[email protected]

[fyellin]

@logophile: Sorry, I didn't word that well. Scalene triangles with integral area and right triangles with integral hypotenuses are rather sparse. Add in the condition that they are primitive ( gcd(a,b,c)=1) and they are even rarer. Given a clue with 100,000 remaining possible ways of filling it in, it was easy enough to look at all of them to see which were a possible integral solution. I'd often get only 1 or 2.

(I sure hope this is considered discussing math and general solving techniques, and not giving hints. I'm not talking about the answer to any particular clue.)

[midazolam]

Clamzy I am with you. I finished this without the use of Excel etc to create formulae (probably because I can't!). It would be nice to know how you can put a formula into excel to give you a long list of possibilities. I did however use a few online calculators to speed things up (e.g. a factor calculator)

The thematic link between the rows and columns came to me at work and therefore if one uses a bit of logic and lateral thinking keeping in mind the preamble does say "thematic" it is gettable without the recourse to mysterons advice.

I am getting annoyed at this site's continuing disregard of the use of hints and it is a shame that given Elap has personally asked for decorum there are those that fail to use their grey matter and incessantly ask, and those that give too much away.

Thank you Elap for probably the hardest numerical in the listener for some time, and the hardest in my time solving, with an extremely impressive construction both in terms of grid and clue progression (and that is form a numberphobe).

[email protected]

[silversolver]

Came late to this one and found it pretty hard work. The numericals are generally not my favourites although they do add some welcome variety. I managed to fill in the grid (largely through brute force and trial and error rather than through any mathematical skill). The final step eluded me for quite a while but the penny has finally dropped.

I agree with midazolam that some of the hints on this site are in danger of becoming a bit too blatant,

[daagg]

midazolam, I use Excel to generate a table of possibilities for each formula. I always use Excel in the "old" mode with cells labelled A1 etc. For example, to generate a table of x*y:

B1=1
C1=B1+1 and copy right as far as you like
A2=1
A3=A2+1 and copy down as far as you like
B2=$A2*B$1 and copy right and and then down to fill in the table

The $ symbols stop the source row and/or column moving with the copy.

[jamesah]

Finally understood thanks to some valuable help! Unfortunately my grey matter would not have helped as I have never heard of the term used to describe the numbers concerned. I don't think that I will get the final few squares as my calculator can't manage 11 digit numbers. Back to normal on Friday I am glad to say.

[sohcahtoa]

Can I check if anyone agrees with 693 for E3? I seem to have come to a halt further on so would like to check on one clue.

[kwyjibo]

I started this on Sunday, went wrong somewhere and if i hadn't been snowed in may not have tried again. I went right back to the beginning and found that I had been on the right lines but made a silly error. On reflection it probably isn't as diificult a puzzle as I made it, although you need a couple of formulas and a spreadsheet to fill in the missing corners.

[kwyjibo]

sohcatoa - E3 is not 693.

[Mysterons]

jamesah - try Start - Programs - Accessories on your PC - there should be a calculator there. No real need for spreadsheets to complete the corners if you use the formula to find the roots.

[dr b]

100!!!! Hoooooooooo!!!!


Sorry.