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!

[AndrewG-S]

Your wide? Not very flattering.

[AHearer]

Oops, don't tell her I said that!

[tweaker]

BLACKHUGH

Could you email me please. I left you hanging on an email query late summertime. Sorry. I lost your email.

[email protected]

Cheers

[starwalker]

I have finally reached the stage where only the shaded cells remain. And there, for the moment at least, I am stuck. I cannot remember ever completing a maths puzzle as tough as this.

[daagg]

According to the web, a primitive triangle is one where the three sides have no factor in common greater than 1. Thus, for example, (3, 3, 4) is a primitive triangle, even though two sides are equal. I think the definition in the preamble is a bit ambiguous.

Anyway, I have had a full grid bar the pink triangles since 7pm on Friday, and I still have absolutely no idea what to do next. Even the Saturday night wine failed to bring enlightenment!

[Jogler]

Snap, Daagg!

[Eril]

daagg, (3,3,4) is not primitive because the first two sides both havea factor of 3.

[dr b]

(3, 3, 4) is primitive, since *all 3 sides* must share a common denominator other than 1 for it to be "not primitive".

[Mysterons]

Not quite dr b - no two sides must share a factor greater than 1:

http://mathworld.wolf...iveRightTriangle.html

[daagg]

Eril, that is why I think the preamble is ambiguous. It says "the sides are relatively prime", whereas I think it should say "the sides have no factor in common greater than 1". This matters to me because I have one triangle which has a pair of sides with a common factor, and I am continually worrying that I should start again. On the other hand, too many things have matched for me to have made a mistake.

Of course, since I haven't finished, I might be completely wrong :-)

[daagg]

Mysterons, your link says that gcd(a,b,c)=1. This allows gcd(a,b)>1 or gcd(a,c)>1 or gcd(b,c)>1, or even all three of these.

[Mysterons]

It states that the greatest divisor common to any of the sides is 1 - this is the case for the nine triangles in this puzzle.

[BlackHugh]

Well - I managed to fill in the grid (obviously excluding the shaded squares, which I have not done yet) without worrying about the relative "prime-ness" or not of the triangles. So don't fret!

[dr b]

Finally got back to the shaded squares, and have them filled in. Definitely a tough numerical!

I managed to stay awake for Harry Potter 7a (it was a matinee and I'd had coffee), but honestly, if I want to see hormonal teenagers alternately acting moody and doing inexplicable things at high volume, I don't really need to leave home.

[Eril]

Oops, you are quite right Daagg, I have found the triangle you mean. Agreed too that the actual definition of a primitive triangle is the one you have given (i.e. (3,3,4) is indeed primitive). Well spotted, sorry about that. Just harking back to last week - did you decide to put in that hyphenated word despite it not being in Chambers? If it is not allowed then one of the final elements doesn't look quite right, but would still be identifiable. Have made a decision on it and put the xword in the post, but still would appreciate your opinion.

[Alekhine]

Maybe numbers are my thing, but I loved this and found it quite easy.

Quite a bit of number-crunching in Excel, though, which is not fun for all.

[Mysterons]

Eril, I think the preamble means that the lengths of the triangle sides are pairwise relatively prime, so (3,3,4) would not be primitive as the first pair of sides share the factor 3 - as BlackHugh says, it's fairy academic in the context of this puzzle.

[Rogue-Elfe]

Hi guys. I'm like a few people have been on this post so far: All there bar the three shaded bits. Firstly, everyone, relatively prime means, essentially, if one of the sides has a factor (greater than 1, obviously), then neither of the other two sides have that factor. Another way of saying it is that any two of the sides are coprime. Evidently, if all three of the sides are prime numbers, then it goes without saying it's right. However, notice for the right angles that one of them is of the form 2pq so cannot possibly be prime. (By the way, I didn't use that formula at all in my solving).

Having got that hopefully unambiguous for everyone; I need inspiration as to what is going on in every row and every column. I've tried the following, all of which it isn't:
square numbers, cubes, divisible by 3, products of square numbers, products of cubes. AndrewG-S, dr b, if you are still online - just a tad of a hint of a clue please?

[oldvic]

Please dontr give any hints in this site.

If you do, you will be stripped naked, hung upside down, and then beaten on the soles of your feet with a dead carp.

So there.

Elap

[daagg]

Votes for the definition of "primitive" as used in this puzzle seem to be about 50:50 between:

1) gcd(a, b)=1, gcd(a, c)=1 and gcd(b,c)=1
2) gcd(a, b, c)=1

If (1) is correct, then I should start again. If (2) is correct, then I can carry on banging my head against the wall.