bonanova

I have an even easier one: any prime number can be written as n. Unfortunately I can't complete the proof. Musings ... The proof works for any (3p-1); I wonder why the puzzle writer chose 26. 299999 (300000-1) might be discouraging? -1 trivial? ["prime = n" works there.]

I'll add the words all and equally likely to PDR's nice explanation. And echo Martini's caveat that the puzzle writer's attempt to fool you often depends on your answering a question you presume is being asked. Read it carefully. For example, They have two kids, one of them is a girl, what is the probability that the other kid is also a girl. All equally likely possibilities = GG, GB, BG. Favorables = GG. Answer = 1/3. They have two kids, the oldest is a girl, what is the probability that the other kid is also a girl. All equally likely possibilities = GG, GB. Favorables = GG. Answer = 1/2. The possibilities must be equally likely; otherwise ... I just bought a lottery ticket. What's the probability I will win? All possibilities: It's a winning ticket, it's a losing ticket. Favorables: it's a winning ticket. Answer = 1/2

Can't claim credit. I'm quoting it from a long forgotten source.

And to you! Cute.

One of my favs ... Sir, I bear a rhyme excelling In mystic force and magic spelling Celestial sprites elucidate All my own striving can't relate

I can't make out the 2nd and 4th [same] object in the bottom line. Can we get a clue on that? Very seasonal code ... ! * he says, as his thoughts turn mysteriously toward his refrigerator *

Let the height - apex to base - of the original pyramid be c. Cut the pyramid with a plane parallel to the original base a distance b from the apex. This defines a second pyramid with height b. Cut again, a distance a from the apex. This defines a third pyramid, of height a. Volume of any pyramid is proportional to the product of base area and height [1/3 i think - doesn't matter] Area of pyramid bases is proportional to the square of their heights So, volume of pyramids is proportional to cube of their heights. You want the pyramid volumes to be in the ratio of 1[a] : 2(B) : 3[c] so that the slices will have equal volumes. a[cubed] = 1/2 b[cubed] = 1/3 c[cubed]. That should give you the ratios you need. Caveat, I did this on the fly, and I'm only half awake, so maybe some other genius will come along and correct this.

What does "they are both on the opposite side of the bridge" mean? If it means they are together, on the opposite side of the bridge, then If it means they are apart, on opposite sides of the bridge, then

mgrannis -- will we get the solution? Or was this a drive-by puzzling?

The proof works anyway, doesn't it? [primes are odd.] Proving (a*a) + 26 is not prime whenever a is odd may be shooting a rabbit with an elephant gun, but the rabbit bites the dust regardless.

#1 said the other alien was both a Huggle and a liar. Impossible. That reveals #1 as the liar. The guidance from #1 that the other alien is a huggle dorfid is not reliable. As I noted in previous post: If we take #1's last statements to be a logical argument [because A is true, you must therefore do B] then we can analyze the argument if we want, and in this case find we'd find the argument is not valid. Then we might conclude that a truth-teller cannot state an invalid argument. Two problems with that: [1] I don't think #1's last statements are an argument. He made 2 independent statements. [2] Stating an invalid argument may not be the same as lying. The speaker would have to have further said ... It is valid to conclude that because A is true, you must therefore do B. That statement would be a lie. But those words are not in the OP. Liars and truth-tellers are more reliable revealed by [a] the truth values of their statements than by the validity of their arguments.

I missed that conclusion in my solution. I took all the statements as red herrings except for the last one. I guess I concluded that both of them would accuse the other, and so nothing could be made of that. It's an interesting puzzle, and It took me awhile to understand it.

I was unaware of writersblock's 6n+/-1 formulas, but it looks as if it's true for any odd a, thus for any 6n=/-1 and for any prime. I wonder why 26 was chosen in the initial problem. It seems that 26 could be replaced by any number that's 1 less than a multiple of 3. For example, -1. But that would be trivial: [a*a]-1 is obviously even for any odd a. So maybe 26 is just a number that looks arbitrary and would seem difficult to include in a proof. Nice job, both!

Nice one. I actually didn't get that far in my analysis.

Thanks. I must have deleted the end tag on an edit. For me, maybe you should disable after zero minutes...

Well I thought about using tim and avrag, but ... Nice job!

It might be harder than that. If 26 were a multiple, but it's only added. I'm at a loss on this one.

I'm awestruck! All the best...!

Business is getting so bad -- last month the Mafia laid off 50 judges.

It's coming, you know. And so unusual. Santa, snow, and so on. This paragraph is unusual, too. How quickly can you find out what is so uncommon about it? It looks so ordinary that you may thing nothing is odd about it until you match it with most paragraphs this long. If you put your mind to it, and study it, you will find out, but nobody may assist you; do it without any coaching. Go to work and try your skill at figuring it out. Par on it is about half an hour. Good luck -- don't blow your cool.