pouzzler
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A little known fact of Beethro's life, is that, before entering Delving School (I'm sorry, but this account is taken straight from The Lyves and Facktes of Beethrough Boodkin, a book pilfered by a brave adventurer in the Empire's deepest vaults - hence the occasional unsavory terms or comments), Beethro followed a cursus as a mathematician at King Dugan's Unyversity of Hygh Mathematicks - the same Unyversity Halph, a precocious lad, was attending before the tragedy-prone summer vacations during which awful events occured which are related in "Journey to Rooted Hold".

The King of All Goblins, having laid his grimy green hands on them, knowing this, decided to award Beethro and Halph a "last chance" before, I quote, "tastings their meatss".

"..hhhrmmm," he goes, clearing his throat while approaching the two his eyes on his notes. "It so happens, that I have a little problem to submit to you. Solve it, and I will reluctantly let you go free, fail and I WILL TASTES YOUR MEATS. BWAHAHAHAHAHA!"

"I chose two numbers between two and twenty, inclusive", he rants on.
"On this here slip of paper," he says, giving it to Beethro, "is their sum."
"And on this here other slip of paper," he adds, giving it to Halph, "is their product."

"And now...", beaming a truly demoniac grin, he says, "What are the two numbers? BWAHAHAHAHAHAHA!"


Halph says: "I can't tell, Unka."
Beethro answers: "I knew that, pumpkin."
To the Goblin King's surprise, Halph instantly answers: "Then I know both numbers, Beethro!", to which Beethro retorts: "Really? Me too, then."

Beethro then gets his sword out and carves both numbers on the slip of paper given to him by the Goblin King, while Halph does the same with a more modest pen he gets out of his pocket. They then hand out the slips to the King, who furious, red with barely unrepressed anger, expels them out of his hands, hold, and dominions through a nasty chain of epithets.


What are those numbers?

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[Last edited by pouzzler at 07-12-2005 12:23 PM]

Maurog
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I have a problem with that problem.


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eytanz
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Maurog:

Well, I have the same problem, but note that you made a mistake in the following:



In fact,



But that just makes the issue worse, because Halph could have gotten:


And I can't see how Beethro would figure out which.

Maurog
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Sorry to spoil it for you, but I made no mistake.


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Banjooie
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These are the logical fallacies that confuse me.




So, yeah.

TripleM
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I have come to exactly the same conclusion as Maurog. Theres no way for Beethro to know which of the 4 possibilities it is.

eytanz
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Maurog wrote:
Sorry to spoil it for you, but I made no mistake.

Click here to view the secret text

×17 is broken into 11 and 6 which give 66, and are the only two numbers in the range 2-20 to give 66. 23 of course gives us 19 and 4, for a total of 76. You cannot make a 76 of any other two numbers in the range 2-20.
I repeat, 11 is the only number in the range 4-40 that cannot be broken into two numbers that disclose themselves in the product.

Oh, yeah. There was a bug in the program I wrote to solve this... Sorry.

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pouzzler
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If I am not mistaken, the sum 12 could be, possibly, 2+10, and 2x10=20, and 20 could possibly be 2x10, or 4x5 - so I think 12 also belongs to the numbers that can't be broken in numbers whose product instantly reveals them; right?
I'm beginning to get a little confused too.

I have the solution in a book, but that would be cheating, so I promise to look at it on my own before retorting to cheating, and that will be as of tomorrow, because four-day week-end is coming then.

VortexSurfer
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pouzzler wrote:
If I am not mistaken, the sum 12 could be, possibly, 2+10, and 2x10=20, and 20 could possibly be 2x10, or 4x5

That's right, but the point is that 12 could also be 5 + 7, in which case the product would instantly reveal these numbers. Therefore, if Beethro gets 12, he cannot know that Halph cannot know the answer.

Maurog
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It's only 11

You can forget about all the even numbers because they all can be represented by a sum of two primes (known theorem). You can forget about all odd numbers that are a prime+2. This leaves several other odd numbers and I checked them all. 11 is the only option.

And we are not even close to explaining "pumpkin"...

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KevG
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Maurog wrote:This leaves several other odd numbers and I checked them all.

You actually don't even need to check them. If the numbers are 11 and X, then the product will instantly reveal that fact since one factor must be a multiple of 11 and only 11 itself is smaller than 20. This eliminates everything from 13 to 31. Substituting 19 for 11 eliminates 32 through 40.

11 is the only option.

Agreed. My guess would be that the problem is copied incorrectly.

rothro
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Hi.

KevG wrote:

Maurog wrote:This leaves several other odd numbers and I checked them all.

You actually don't even need to check them. If the numbers are 11 and X, then the product will instantly reveal that fact since one factor must be a multiple of 11 and only 11 itself is smaller than 20. This eliminates everything from 13 to 31. Substituting 19 for 11 eliminates 32 through 40.

11 is the only option.

Agreed. My guess would be that the problem is copied incorrectly.

So far I agree with you all, and I don't see a unique solution for Beethro either, UNLESS King Dugan's Unyversity of Hygh Mathematicks is entered as a vital clue. (This leaves the realm of logic and enters that of wild guesses.) You see, in that university mathematics course three maximes are taught over and over again:

1) Approach a problem in an orderly fashion and work your way up.

2) The first answer that springs to mind is usually the correct one.

3) If several answers appear equally valid chose one, declare a schism, and smite the supporters of the other solutions.

Ok, the third one may have been a joke perpetuated by students who got tired of hearing the first two over and over again. But now we know that Beethro knows that Halph knows these rules. Thus only remain as valid options.

Tada...
Rothro



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Maurog
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So what you're saying is that even though Halph got 30 on his little note he answered 2 and 9?

This either makes lots of sense, makes no sense at all, or is a purple penguin.

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rothro
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Maurog wrote:
So what you're saying is that even though Halph got 30 on his little note he answered 2 and 9?

This either makes lots of sense, makes no sense at all, or is a purple penguin.

No, he clearly had 18 on his note... this depends on how you define "work your way up orderly". I assumed since there were two finite sets I'd order them (2,2) (2,3) (2,4) ... (2,20) (3,2) ... (20,20).
Of course you can chose (2,2) (3,2) (2,3) (2,4) (3,3) (4,2) (5,2) etc (like in the old proof that the rational numbers are countable) in which case Halph gets 30, the numbers are 5 and 6 and my purple penguin turns into a blue moose.

Anyway - this approach was not meant seriously

Cheers,
Rothro.

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Maurog
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What I meant is, the number Halph got has nothing to do with how you order the sets and everything to do with what numbers the Goblin King had in mind. And no matter what rules Halph is going by, he knows the numbers but Beethro doesn't.

Now, we all know Beethro cheated on this one, but how?

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trick
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Maurog wrote:
Now, we all know Beethro cheated on this one, but how?

The first post clearly states that Beethro used his sword to carve the answer. Thanks to careful sword control, Beethro must have been able to see the reflection of Halphs paper in his sword when he whipped it out, and so were able to write the correct answer himself.

- Gerry

pouzzler
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I humbly beg everyone's pardon.
The original problem went like that:

".... two numbers between two and a hundred; as this is a difficult problem, try your luck beginning with two numbers between two and twenty."

It seems the 2-20 hypothesis had been insufficiently researched; however the problem is valid for 2-100 as I have checked the solution.

*holds bare back for whipping*

Banjooie
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I still demand answers as to this 'pumpkin' and 'Unka' thing, since you claimed they were colleagues at a university.

pouzzler
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Oh! Not so fast, nimble instrument of musicoie!
I merely claimed they attended the same university! Not necessarily at the same time.

"Pumpkin" seemed to me to be a word used when speaking to one's youngsters, conveying affection - yet, my knowledge of the english tongue being purely (not necessarily in that order)
academic
Buffy vs. the Vampires-ic
Science-Fictionic
heroïc-Fantasiac

I may be wrong. If so, I may extend lower parts of my back for whipping, but not necessarily.

TripleM
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Well, I used computer assistance, so I'll let someone else think of a way of explaining things nicely, but the answer, I think:


Method:

Banjooie
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pouzzler wrote:
Oh! Not so fast, nimble instrument of musicoie!
I merely claimed they attended the same university! Not necessarily at the same time.
...I may be wrong. If so, I may extend lower parts of my back for whipping, but not necessarily.

Then let us 'delve' into the beginning of the disturbing parts of your argument.

A little known fact of Beethro's life, is that, before entering Delving School..

Assume we do not take the Beethro's First Job winning story as canon. (As it rightfully should be.)

Beethro cannot possibly be more than 40. He is /certainly/ middle aged, and we know he has had 'decades' of experience. For the sake of argument, I will go with '20 years'.

Meaning, he went into this university at 20. But wait, where's Halph?

Halph is not twenty or older.

Therefore, it must be a different Halph entirely, and not one that has Beethro for an uncle.

Furthermore, even if it were somehow the real Halph, Beethro has never referred to /any/ of his nephews with anything even remotely approaching affection.

Yeah, you people go deal with your 'math'. I've got more important things to quibble and quarrel about!

pouzzler
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I will quote myself, even though it reeks of arrogance

.."Halph, a precocious lad, "...

I tell you this sonny, jus' between you and me, I ain't going down for trying to give a Droddy flavour to some problem I stole from old piles of math magazines I have lying around... or if I do, I'll take the Eigth with me, bwahaha!


PS: Oh by the way, is that really so?
Hereabouts, university begins (normally) at eighteen for the lucky (not so) few.

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[Last edited by pouzzler at 07-15-2005 08:33 PM]

Banjooie
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--Okay, what I'm saying is that Halph is not 20 at the time of JtRH, when we see him.

He is maybe 12.

This means, that in the 20 years ago or so that Beethro would be in university, Halph would be roughly -8 years old.

That's pretty precocious.

agaricus5
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Banjooie wrote:
--Okay, what I'm saying is that Halph is not 20 at the time of JtRH, when we see him.

He is maybe 12.

This means, that in the 20 years ago or so that Beethro would be in university, Halph would be roughly -8 years old.

That's pretty precocious.

The puzzle itself doesn't actually mention what Halph was doing at the university, though. Possibly he was attending some sort of "masterclass" intended for younger students, or even was receiving some specialist training there that it wasn't possible to get elsewhere. However, I am speculating, so I could be totally wrong. Also, the credibility of such an old book should be questioned (its translation into Olde Syhmpoohl Enhklyhshe suggests it is rather ancient), since before "the new age of reform and thought" (Post-B.D.) records from the Eighth were not catalogued or maintained by the Eighther Information Archives (E.I.A.) and so it cannot be confirmed whether or not such events ever happened (with comparison to other records).

Of course, Beethro and Halph were not alive at the time when such writing was in use, unless the place in the Empire where it was written was particularly backwards, so it may be a completely different Beethro and Halph that is referred to.

Possibly.

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[Last edited by agaricus5 at 07-16-2005 09:48 PM]

Banjooie
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The puzzle itself doesn't actually mention what Halph was doing at the university, though. Possibly he was attending some sort of "masterclass" intended for younger students...

What kind of masterclass is intended for foetuses that won't be conceived for another 8 years? :O

Of course, Beethro and Halph were not alive at the time when such writing was in use, unless the place in the Empire where it was written was particularly backwards, so it may be a completely different Beethro and Halph that is referred to.

True, true. I can't really argue this point, but I am still vaguely disturbed by this 'pumpkin' comment.

eytanz
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It's my understanding that "pumpkin" is a term of endearment used for girls, not boys. That may be what's bothering some people...

Halph in the game is definitely younger than 20 but he's also older than 12 - he's about 14, I think (I don't remember if Erik ever gave us an official age, but he specifically mentioned that he's a teenager several times, not a pre-teen).

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rowrow
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I thought I remember someone mentioning that he was 13. I'll look for it.

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agaricus5
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It suggests in the ending that Halph is 13:

[From text.dat (which, incidentally appears very scrambled)]

Ten-year-old Tyler is about three years younger than Halph, whom he plays. He gave the character an authentic childish quality that definitely came through.

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