Indeed, 46464, 69696 are the only possible positive decimal palindromes that satisfy the given conditions.

However, I do not know of an easy analytic method (and still looking for it) to substantiate this.

Notes:

1. It may be observed that a palindrome is always an integer according to this mathworld article , unless stated otherwise.

2. According to wikipedia, decimal denotes base ten.

In hindsight, I should have mentioned “base ten representation……” in the problem text rather than “decimal representation”.

I wrote a simple little program to solve this problem... And I knew the upper bound for the number was 70710, since any number greater than that would have more than 10 digits in the resulting answer (which invalidates the requirement that all numbers be present in answer)...same with the lower bound being 22360 (since the resulting number would have less than 10 digits). See the attached code:

List<String> palindromes = getPalindromes(22360, 70710);
for (int i = 0; i < palindromes.size(); i++) {
Long l = Long.valueOf(palindromes.get(i));
Long newVal = 2 * (l * l);
String newStr = newVal.toString();
boolean hasAll = true;
for (int j = 0; j < 10; j++) {
if (newStr.indexOf(Integer.toString(j)) < 0) {
hasAll = false;
break;
}
}
if (hasAll) {
System.out.println(l + ": (2 * " + l + " * " + l + ") = " + newStr);
}
}
}

private static List<String> getPalindromes(long lowerBound, long upperBound) {
List<String> palindromes = new ArrayList<String>();
for (long i = lowerBound; i <= upperBound; i++) {
String val = Long.toString(i);
boolean isPal = true;
for (int j = 0; j < val.length(); j++) {
if (val.charAt(j) != val.charAt(val.length() - (j + 1))) {
isPal = false;
break;
}
}
if (isPal) {
palindromes.add(val);
}
}
return palindromes;
}

    public static void main(final String[] args) {

Basically just get all numbers that are palindromes between those bounds, and then go through each of those palindromes, perform the operation, and check to make sure each resulting answer has every number. Fun problem, I enjoyed it.

Determine all possible positive palindrome(s) N, such that the decimal representation of 2N² has no leading zeroes and contains each of the digits from 0 to 9 exactly once.

Here are the only 2 INTEGER solutions to this problem...I'm sure there are some decimal answers as well...

46464: 2(46464)² = 4317806592

69696: 2(69696)² = 9715064832

Which then you can just add decimal places to, to get some REAL solutions:

4.6464

46.464

464.64

4646.4

6.9696

69.696

696.96

6969.6...So, I will guess that that is all of them, (since the result of .46464 and .69696 have leading zeros in the answer.

A grandfather is telling his grandson war stories.

"At the end of World War 1, I was awarded for my bravery after saving a group of my men," the grandfather says. "You see, we were fighting in northern France and one of our enemies threw a grenade at us. I managed to pick it up and throw it away before it exploded. So right after the war ended, a General gave me a sword, engraved with the words 'Awarded for Bravery and Valor, A True Hero, World War 1'."

The grandson thinks about the story for a minute and then says "Grandpa, that story can't be true!" How did the grandson know?

It's just like the riddle about an archeologist trying to sell some of his artifacts to a museum, and he claimed he found artifacts from around 500 B.C. including pots, tools, some artwork, and money from the time. The way he knows that it's from around 500 B.C. is because everything was made using materials available at the time, the money was coins that had years 530 B.C. - 490 B.C. on them, and the artwork was of events that happened at the time. The museum curator immediately threw the man out and called him a fraud...why?

(Not trying to start a new discussion, just saying that the reasoning is the same)

A grandfather is telling his grandson war stories.

"At the end of World War 1, I was awarded for my bravery after saving a group of my men," the grandfather says. "You see, we were fighting in northern France and one of our enemies threw a grenade at us. I managed to pick it up and throw it away before it exploded. So right after the war ended, a General gave me a sword, engraved with the words 'Awarded for Bravery and Valor, A True Hero, World War 1'."

The grandson thinks about the story for a minute and then says "Grandpa, that story can't be true!" How did the grandson know?

It wouldn't have said "World War I" since at the time it was called "The Great War"...because they had no way of knowing there would be more than 1

[Binary]

Binary: 2011000103011000013011100113011010013011000112

Hint: The solution is in lowercase.

*ok seen as this has been here for a little while and no one has got it yet, I might as well put the solution in*

This is a misdirection puzzle, the 2's & 3's are there to throw you off, get rid of the 2's & 3's and that leaves you with (Binary) to decode.

I wont print the decoded text here, I will leave that up to you

Again, i saw this one that has been "dead" for a few years...but no one gave the answer to it...and it's unfortunate that this user hasn't had a post in a year and a half (and this was their only one) and that no one solved it...because that probably means they won't verify the answer, but believe me, it's right...

So "BASIC" to me...yay ASCII!!

So, I learned this one when I was in my Calculus class and thought it was just sort of fun. Basically, there is a shape, called Gabriel's Horn (where you take the curve generated by the function f(x) = 1/x (where x >= 1)) and revolve it around the x-axis to form a horn shape.

Using calculus, you can calculate the volume of this shape (it doesn't show up very well, but π is PI:

Volume = π ₁∫^a (1/x²) dx = π (1 – 1/a)

So, lim_a->∞ π (1 – 1/a) = π (1 – 0) = π

The surface area can also be calculate as:

Surface Area = 2π ₁∫^a √(1 + 1/x⁴) / x dx

We know that √(1 + 1/x⁴) > √(1)...so we know that the Surface Area > 2π ₁∫^a √(1) / x dx = 2π ln a

So, lim_a->∞ 2π ln a = ∞

You might ask why this is a paradox...you can think of it this way:

Imagine you are a painter, it would take infinite amount of paint to paint the surface of this figure...however, you could FILL the entire shape with π volume of paint.

I like this one (I know WHY it isn't a paradox...but that's no fun to give that away right away...), so I thought I'd post it.

the waiter did take the fly out.

Sorry, I had changed my answer before you posted this reply...so, maybe my new answer is correct?

Can you find the explanation for the following?

Snoop and Watson had just enjoyed an excellent meal (chips in a basket) in their favourite restaurant (the Goat and Compasses). Having partaken liberally, not to say magnificently, of beers, wines and spirits with their gourmet fare (or fayre, as the establishment had it), they decided to round off the occasion with a cup of black coffee each.

"I say, Snoop," said Watson when Snoop, after stirring it, began to sip his coffee with the air of one unused to enjoying such sobering refreshments, "isn't that a pair of bluebottles in your cup?"

Snoop spluttered, discharging a fine mist of coffee into various parts of the neighbourhood. "Not a pair, Watson," he then pronounced, "but a single - not to say singularly large specimen of bluebottlia restauratis. Waiter!"

Snoop explained the situation to the waiter and (having got through the obligatory rigmarole about everyone wanting one) his cup was removed and a new one brought, minus fly.

It was definitely a different cup, and certainly free of fauna, but on taking a sip, Snoop spluttered again. "Pah!" he said, "this is the same coffee I had before - only the cup has been changed!"

"How do you know?" asked Watson, "is it cold?"

"It doesn't seem any colder than it was before," replied Snoop, "but it's definitely the same coffee."

"How can you be so sure?" Watson goggled, "after all, cheap coffee's cheap coffee."

"Ah, but there is one thing you've forgotten..." began Snoop.

What had Watson forgotten, and how could Snoop be so sure it was the same coffee in a different cup?

I'm going to say that since he was stirring his coffee, and he wasn't used to such sobering refreshments, that he had mixed something into his coffee (like cream and/or sugar), and so he could tell it was still his coffee when he got it back...

[Replicant]

what numbers can you put at least 2 different mathematical symbol between them (1 at a time)and the result is the same?

give at least two nos.

This riddle has been quiet for a few years...but I just saw it.

I would say that there are infinite numbers that work for this...All positive numbers can simply be:

ⁿ/_n = 1

_nC_n = 1

Since any number divided by itself = 1, and any number CHOOSE itself = 1

There are five people in a room. Each person is either a knight, who always tells the truth, or a liar, who always lies. Each person is asked the following question: How many liars are among you? The answers are: "one", "two", "three", "four", "five".

How many liars are in the room?

I agree with the above that there are 4 liars, 1 knight.

If there were 0 Knights, then the liar who said there were 5 liars would be telling the truth, so that doesn't work...and if there were more than 1 knight, than 2 answers would have been the same... So, the person who said "four" is your knight, the rest are liars.

even with the math i still do not understand how you would reach a different point from which you start if you are traveling in a square? Assuming the math is right of course. I still dont understand how this riddle is possible.

Since no one has really answered your question on how this is possible...it's simply because you're thinking in Euclidean Geometry (which is understandable, since that's what we are all taught from grade school), but since we are talking about the surface of the earth (assuming a perfect sphere)...we have to use

Spherical Geometry (this link has a good image showing a triangle in spherical geometry with sum of angles greater than 180 degrees...which is impossible in Euclidean Geometry). In short, you are not travelling in a "square" so to speak...you are traveling south (assuming a meaning of heading directly towards the south pole) around a curved surface 400mi, then east (assuming parallell to the equator) around it 400mi, then north (again, assuming directly towards the north pole) 400mi, and because of the curvature of the surface, you are actually more than 400mi to the east of your original starting point.

It's very similar to the old riddle saying, you're somewhere on the earth and you travel south 100 miles, then east 100 miles, and finally north 100 miles...and somehow you ended up exactly where you started...how is this possible. The answer is you are at the north pole (use the picture on that link above to a visual reference)

Oh, how I miss my non-Euclidean Geometry class...Hope this helped.

it is old but also new,

i want it buried but if someone tries they can drag it up.

its something that can harm me but will help them.

people judge me for it, while some want me dead for it.

i've disapeared but it is said i'll come back.

i'm not tupac, i am like the string killer.

Something like innocence? Not sure about the zodiac clue....

New puzzle.

534344334345454535_ _ _

What are the next three numbers.

well, you messed up one part of it...it shouldn't be:

534344334345454535...

it should be:

534344334545454535...

But regardless, the next numbers would be 543

I don't know about you guys, but I certainly can foresee my little differences, as well as my big differences. Madam Edelman has a point, though

[Riddle of the Species]

No,

But it is a type of colorful fish.

Rainbow...so, rainbow fish...

umm...

how many windows are in this place? I mean, who cares if the lights go out...since it's lunch time, if there are ANY windows, then SOMEONE must have seen something, right?

It's a pretty straight forward cryptogram (depending on which level). Decode the message, each letter is consistently-(always) replaced-(maybe) by a different letter in the alphabet.

And don't forget the spoilers!

-This puzzle is what I call a level - 2

Which means that this puzzle is - A short quote cryptogram. Possessing no conjugations or double letters. Also includes all letters and the name of the quoted person.

Good Luck!

Jc uzphz eaz vlcv ynmx eazjm ipeazmv, yne jc fpm eaz ipeazmv ynmx eazjm vlcv. ~Hmlzvnv

For some reason this one just came to me...not usually that great at these:

In peace the sons bury their fathers, but in war the fathers bury their sons. ~Croesus

Continuing on from the good work started by Gubdude:

1. By the end of the __________, I feel ________

2. It looks like a _______, but it cannot ______

3. Bird ________ were the ________ of the noise

4. After the renovation he noticed _______ in the ________ (plural)

5. The interior of the _______ was very ________

6. His efforts at _______ the leak in the ________ failed.

7. It was her ________ that ________ him reeling

8. He only posted on the _______ when he was _______

2: Bee/Be

3: Caws/Cause

8: Board/Bored

What tgcat was implying is once the coins are stacked they become a column. You now have 2 rows of 3 coins, 4 columns 1 coin high and 1 column 2 coins high.

he meant that:

C C C is a ROW

while

C

C

C

is a column...since rows go left to right, and columns go up and down...that's another explanation of his statement... in which case the solution of

c C c
c

  c  

has 1 ROW of 4 coins, and 1 COLUMN of 4 coins...

Never seen, Never heard,

I am what is in this world,

Something always neglected,

Someone never blammed.

Thru space and time i make my way,

With the hope to one day,

Find an equal to hold my hand,

'Till the time they pass me by.

I am one to make all bliss,

Worth living and dying for.

I am the one that will someday,

Take you and more to a place of love and hate.

I am what i am. And I, will forever be.

As for you, i will never leave,

Always standing closeby,

From your begining and at the end.

Not very convinced myself about this, but i hope some can have some fun :-)

Death or Grim Reaper or whatever symbol you want for Death

Some of us perform in circus shows

How deep we dig, it's hard to know

If you need a house, give us a call

For some of us are builders, after all

Our army numbers in the billions

We swarm our enemies, even children

We have bullets but no guns

We breathe fire like the sun

Not strictly scavengers, we can farm, if you please

Growing crops using leaves

So now you know some about what we do

Who can tell me what we are? Can you?

Ants...based on the key words "builders", "digging", "fire", "farm", "billions", "army", and just the over all feel...

Yep, another one right. Congratulations!! Here's your free virtual zoo ticket. Just imagine you're in a zoo. By the way, was this too easy?

In my opinion, it was too easy. I would say that it was just a basic geometry problem (so, I would guess around 8th/9th grade math level, depending on where you go to school). So, it may have been a good challenge/exercise for some people on here...To make it a bit harder, you could make the hexagons different sizes or irregular, or something else (we could get really crazy with it, but let's just leave it at that)...

Since you say the center of the hexagons are 4 inches away from the corners, I assume you mean away from ALL the corners...in which case they are regular hexagons...

That makes the triangles formed equilateral triangles with sides of length 4 inches...which means that the area of each triangle would be 2 * sqrt(12)...or about 6.93 sq. in.

I was thinking that since the FF has to take 66.666 steps to cover the 200' race and the nurse has to take 100 steps, that the nurse's foot stops exactly at the 200' mark while the FF's foot travels through the finish line. I see it as the FF wins

But, the firefighter can't turn around at the halfway point until he has PASSED the 100' line...he won't do that until he has taken 33.33333 steps...which means he needs to take 34 steps before he can turn around.

The OP says that they are traveling at the same speed (since the firefighter's 2 steps of 3 feet is the same as the nurses 3 steps of 2 feet). The nurse, however, will get to turn around at exactly the 100' mark, while the firefighter has to turn around at the 102' mark. So, the nurse travels exactly 200' during the race, the firefighter travels 204'...

So...since the firefighter has to travel 4' further than the nurse, and they are going the same speed...the nurse will cross just before him.

For each leg of the race, the firefighter has to take 100/3 = 33.33 => 34 steps, or 68 steps total.

For each leg of the race, the nurse has to take 100/2 = 50 steps, or 100 steps total.

She takes 3 steps for every 2 of his, so, because 100/68 < 3/2, the nurse wins.

that's what I was saying in my first post, but this is just a better explanation :c)