Avon
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The solution is:

A) 631
B) 136
C) 233
D) 361
E) 343



[Edited by Avon on 12-31-2003 at 03:12 AM GMT: make article definite]

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Schik
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Though I would have liked to have seen some logic behind it, that is correct. You're up!

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DiMono
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Other than being 3 digits, is there any restriction on the numbers? For instance, do all the digits have to be different?

Edit: Well, that's what I get for not noticing I was on page 8 of 9. Maybe I should stick this as a request on the Site forum...

[Edited by DiMono on 12-31-2003 at 03:52 AM GMT]

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Oneiromancer
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A request for what? Do you also have a Tin Man and a Lion accompanying you?



Sometimes I amuse myself too much.

Game on,

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Avon
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Schik wrote:
Though I would have liked to have seen some logic behind it, that is correct. You're up!

Sorry, I was pressed for time and most of my logic had been erased as I went. I could reconstruct it if you really want me to. But assuming you don't, here is my puzzle.

King Harold had assembled his army, which was divided into regiments. Each of the king's 61 trusted lieutenants commanded a regiment and each regiment contained the same number of men. For tactical reasons, this number was square. When Harold joined the army, they could rearrange themselves to form large square and crush the enemy. What is the smallest posible size for one of King Harold's regiments?

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zex20913
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Are we counting the lieutenants in the big square? Are we counting them in their regiments?

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Avon
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zex20913 wrote:
Are we counting the lieutenants in the big square? Are we counting them in their regiments?

Yes and yes.

The lieutenants were there to ensure each regiment contained at least one man.

Otherwise there would be the trivial solution where the regiments have size zero (a square) and Harold standing with all of his zero men would form a sqaure of size one.

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DiMono
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Dumb question: have you solved this yourself? Because I'm at 35 million per regiment without a solution, and it's getting a little excessive having 2.147 trillion troops.

[Edited by DiMono on 12-31-2003 at 07:52 AM GMT]

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DiMono
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Alright, I've got two computers working on this one, and they've both reached numbers higher than I can count for the regiment sizes and not come up with an answer. Unless the formula's not n*n = (i*i*61)+1 you're going to have to convince me this one's possible.

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Mattcrampy
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Your formula's wrong, DiMono.

Matt

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Mattcrampy
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After pulling my old VB out (my C++ compiler hates me) the answer is:

63 soldiers in each regiment, which join up to make a square 62 soldiers to a side.

I could have looked at the problem and remebered that x^2 -1 = (x+1)(x-1) but the program was cooler.

(Incidently, 2208 soldiers in each regiment also works.)

Now to make a puzzle survive one hour!

Matt

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Schik
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I'm pretty sure you're wrong, Matt.

63*63*61+1 = 242110, and 62*62 = 3844. Not quite equal.

This puzzle brings back not-very-fond memories of Pell's equation and continued fractions.

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Mattcrampy
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Bugger, so I am. It worked on the computer but not on the calculator.

Matt

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zex20913
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Wait a sec, he's right. It's not 63*63*61+1, but only one 63. Unfortunately, we've been told that the number is square, for each regiment. With assumption manipulation, this would work, as well as 59.

So it looks like something is wrong in the assumptions we've been given.

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DiMono
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DiMono wrote:
Alright, I've got two computers working on this one, and they've both reached numbers higher than I can count for the regiment sizes and not come up with an answer. Unless the formula's not n*n = (i*i*61)+1 you're going to have to convince me this one's possible.

I should have been more thorough in my explanation of my equation. n is the side length of the large force, and i is the side length of each regiment. I ran this formula through c code and a perl script, and neither ever gave me an answer. The perl script actually timed out on me and I lost my connection to the server.

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Avon
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I repent of my New Year's Eve wickedness and now that it's New Year's Day I can resolve never to do it again.

Dimono is correct. The numbers involved are excessive. In fact, I have seen a version of this puzzle where you have to say why the story is unbelievable.

And Schick's memories of Pell's equation and continued fractions are probably the key. That's how I solved it when it was a question in a Number Theory assignment, but then we had been taught to do it that way.

Of course, if we changed the 61 regiments to 60 then the smallest possible regiment size would be 16, but that wouldn't be as satisfying to solve.

Happy new year everyone, particularly those of you still back in 2003.

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Mattcrampy
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Okay then, DiMono's up then?

Matt

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The_Red_Hawk
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Are you going to leave it as is, or are you giving us a new puzzle?

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Schik
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Mattcrampy wrote:
Okay then, DiMono's up then?

I don't believe he's come up with a correct answer - I think it's still unsolved. Maybe tomorrow I'll dig out some old math books to remember how to solve this bugger if it's still unsolved.

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agaricus5
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This is as far as I got:

Assuming that the army square length is x, the army size is therefore x^2. Similarily, if the regiment square length is y, the regiment size is y^2.

So...

x^2 = 61y^2 + 1

61y^2 = x^2 - 1

61y^2 = (x - 1)(x + 1)

Since 61 is prime, either (x + 1) or (x - 1) has to be a multiple of 61.

So (x + 1) = 61n,
Or (x - 1) = 61n, where n is some number > 0.

Note: [+/-] stands for plus or minus.

So... 61y^2 = 61n * (61n [+/-] 2)

y^2 = 61n^2 [+/-] 2n.

This means we have to find two square numbers that have a difference of 2n when n^2 is multiplied by 61.

Unfortunately, my Maths does not go beyond that standard, so I'll leave it to someone else to continue it to find out if the solution is impossible or very large.

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Oneiromancer
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Avon wrote:
Dimono is correct. The numbers involved are excessive. In fact, I have seen a version of this puzzle where you have to say why the story is unbelievable.

And Schick's memories of Pell's equation and continued fractions are probably the key. That's how I solved it when it was a question in a Number Theory assignment, but then we had been taught to do it that way.

This is not really a puzzle then; it's more of a math problem put into the language of a puzzle. I don't think that people should be expected to look up Pell's equation to be able to solve a puzzle here.

I think it would be best not to strain everyone's resources and just to let Schik put another puzzle up, since it was his puzzle that Avon solved to give us this new one. Of course if he declines this then everyone can keep chugging away. And I think I will add a new rule that will hopefully preclude these kind of problems in the future.

Game on,

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"He who is certain he knows the ending of things when he is only beginning them is either extremely wise or extremely foolish; no matter which is true, he is certainly an unhappy man, for he has put a knife in the heart of wonder." -- Tad Williams

The_Red_Hawk
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I think that Avon should really be allowed to post another puzzle.

If you don't want him to go again, and if you'll let me, I'd like to put up another puzzle, which I've been "sitting on" for quite a while (I forgot about it last time I posted a puzzle). But maybe that's not right and I should earn it first. Oneiro's decision.

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Slashing, whirling, diving, twirling,
Snapping, turning, rising, swirling,
Screeching, flipping, gliding, sliding,
The red hawk's dance of death.

.....the king of the skies.....

Oneiromancer
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Okay, Avon gets another chance...hopefully he is not too hungover from New Year's and will get us one within a day.
Oh, and tell us the answer to the current one too. Barring, of course, someone else solving it in the meantime.

Game on,

____________________________
"He who is certain he knows the ending of things when he is only beginning them is either extremely wise or extremely foolish; no matter which is true, he is certainly an unhappy man, for he has put a knife in the heart of wonder." -- Tad Williams

Schik
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Oh heck, let me just dust off my memory and solve it to end this odd controversy.

The lowest solution to 61a^2 + 1 = b^2
is a = 226153980, b = 1766319049.

So the regiments have 226153980*226153980 people in them.



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The greatness of a nation and its moral progress can be judged by the way it treats its animals.
--Mahatma Gandhi

Avon
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Okay, now I'm confused. Oneiromancer told me to post another puzzle at the same minute Schik posted a solution. I think I should just sit tight and see what happens next.

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Oneiromancer
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I assumed you would take a long time to get back to the forum since it was already apparently New Year's where you are...ah, New Zealand I see! So it must already be the next morning. Oh well...since Schik solved it you don't have to worry about a new puzzle. Sorry for the confusion. The ball's in his court now.

Game on,

____________________________
"He who is certain he knows the ending of things when he is only beginning them is either extremely wise or extremely foolish; no matter which is true, he is certainly an unhappy man, for he has put a knife in the heart of wonder." -- Tad Williams

Schik
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I'll take the win.

Next puzzle:

You are wearing a blindfold and thick gloves. A very large number of quarters are laid out before you on a gigantic table. Someone tells you that 20 of these quarters are tails and the rest are heads. He says that if you can split the quarters into 2 piles where the number of tails quarters is the same in both piles, then you win all of the quarters. You can never tell what state a quarter is in (the blindfold prevents you from seeing, and the gloves prevent you from feeling which side is heads or tails). How do you partition the quarters so that you can win them all?



____________________________
The greatness of a nation and its moral progress can be judged by the way it treats its animals.
--Mahatma Gandhi

zex20913
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Ah! I have seen this one before. (Just letting you know, so it doesn't get the recognition of a one-hour+ puzzle.)

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levelthirteen
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Simple, stand all the coins on edge.

Schik
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levelthirteen wrote:
Simple, stand all the coins on edge.

Argh. Okay, small addendum to the puzzle:

You have 5 minutes to make your two groups of coins. Since there are a huge number of coins, you cannot stand them all on their edges.


____________________________
The greatness of a nation and its moral progress can be judged by the way it treats its animals.
--Mahatma Gandhi