Solve Enjoyable Enigmas with Mr.E's Challenge

[Ibix]

OK - another dead body, adapted from one of Captain W.E. Johns' Biggles short stories set in the late '40s or early '50s.

A fourteen year old girl leaves school alone on foot at her usual time. She is later found dead, on her usual route home, killed by a single blow to the head. The only habitation close by is a manor house. No one there saw or heard anything suspicious. The only thing out of place at the crime scene is an expensive box of chocolates on the ground nearby. The police are stumped.

At the invitation of Inspector [strike]Lestrade[/strike]Gaskin, Biggles goes to have a look around. While they are at the crime scene, he notices the 20-ish daughter of the family who own the manor house walking up and down the garden holding hands with a man in RAF uniform. Gaskin comments that it was the daughter's birthday the day of the murder, and Biggles immediately suggests arresting the man in uniform. Why?

 

[collinsmark]

OK - another dead body, adapted from one of Captain W.E. Johns' Biggles short stories set in the late '40s or early '50s.

A fourteen year old girl leaves school alone on foot at her usual time. She is later found dead, on her usual route home, killed by a single blow to the head. The only habitation close by is a manor house. No one there saw or heard anything suspicious. The only thing out of place at the crime scene is an expensive box of chocolates on the ground nearby. The police are stumped.

At the invitation of Inspector [strike]Lestrade[/strike]Gaskin, Biggles goes to have a look around. While they are at the crime scene, he notices the 20-ish daughter of the family who own the manor house walking up and down the garden holding hands with a man in RAF uniform. Gaskin comments that it was the daughter's birthday the day of the murder, and Biggles immediately suggests arresting the man in uniform. Why?

I had a dream last night that I solved this. Gwahhh! Why can't I remember the specifics of my dream.

(There's a good chance the solution involves zombies.)

 

[billiards]

OK - another dead body, adapted from one of Captain W.E. Johns' Biggles short stories set in the late '40s or early '50s.

A fourteen year old girl leaves school alone on foot at her usual time. She is later found dead, on her usual route home, killed by a single blow to the head. The only habitation close by is a manor house. No one there saw or heard anything suspicious. The only thing out of place at the crime scene is an expensive box of chocolates on the ground nearby. The police are stumped.

At the invitation of Inspector [strike]Lestrade[/strike]Gaskin, Biggles goes to have a look around. While they are at the crime scene, he notices the 20-ish daughter of the family who own the manor house walking up and down the garden holding hands with a man in RAF uniform. Gaskin comments that it was the daughter's birthday the day of the murder, and Biggles immediately suggests arresting the man in uniform. Why?

Not sure if I've got the hang of these but my guess:

Spoiler

pilot dropped chocolates out of plane on way to his girlfriend's birthday party, hit poor girl on the noggin

 

[Enigman]

Okay thread's lagging...time for a reboot:
So... Chess:
Anand currently is getting his a** kicked¹ and Kasparov's probably grinning ear to ear...
but that has nothing to do with this Enigma.

What is the maximum number of knights that can be arranged on a standard chess board such that not one of them threatens any other or is threatened by any other? ie. no one faces a check.
-----------------------------------------------------------------------------------------
¹ (as of now, if at the time you are reading this it doesn't hold true...I'm a happy man)

 

[billiards]

Okay thread's lagging...time for a reboot:
So... Chess:
Anand currently is getting his a** kicked¹ and Kasparov's probably grinning ear to ear...
but that has nothing to do with this Enigma.

What is the maximum number of knights that can be arranged on a standard chess board such that not one of them threatens any other or is threatened by any other? ie. no one faces a check.
-----------------------------------------------------------------------------------------
¹ (as of now, if at the time you are reading this it doesn't hold true...I'm a happy man)

Spoiler

32. All the dark squares (or all light squares)

 

[Enigman]

Spoiler

32. All the dark squares (or all light squares)

Correct. This one was a different take on the 8 queens puzzle, place 8 queens such that none of them threaten any other, there are more than one solutions. (and if you can do it then try doing it under 43 secs which is my best record...
:superiour irritating smirk:*)

Next one:
Find a number such that it is three times the sum of its digits.
-----------------------
*I need more emoticons

 

[billiards]

Correct. This one was a different take on the 8 queens puzzle, place 8 queens such that none of them threaten any other, there are more than one solutions. (and if you can do it then try doing it under 43 secs which is my best record...
:superiour irritating smirk:*)

Next one:
Find a number such that it is three times the sum of its digits.
-----------------------
*I need more emoticons

Spoiler

27

I'll stop getting them now.

 

[zoobyshoe]

Consider the numbers 43 and 34. We can say one is the reverse of the other.

Consider an isosceles triangle. The two base angles are the same.

It was determined here at PF that there exists a finite number of isosceles triangles whose base angles are the reverse of their apex angle. The total number of these is <10.

Find them and give the angles.

 

[Enigman]

Spoiler

Let triangle be ABC
base angle (<ABC and <ACB)=(100a+10b+c)
<ABC+<ACB >180 if a>1
:.a=0
Base angle=10b+c
20b+2c+10c+b=180
21b+12c=180
7b+4c=60
b has to be even
4,8
8,1
48,84
81,18

 

[zoobyshoe]

Spoiler

Let triangle be ABC
base angle (<ABC and <ACB)=(100a+10b+c)
<ABC+<ACB >180 if a>1
:.a=0
Base angle=10b+c
20b+2c+10c+b=180
21b+12c=180
7b+4c=60
b has to be even
4,8
8,1
48,84
81,18

Correct!

 

[collinsmark]

Here's a good one. Originally this riddle was phrased such that yes or no questions were pretty much essential. I've tried to rephrase it here, putting as much information I can directly into the riddle's wording. I'll still entertain any yes or no questions, but it might not do any good. The essentials are in the riddle. [Edit: well, except perhaps for that Eureka moment.]

Angus, age 34, is found guilty of heinous, heinous crimes by a full jury -- crimes so heinous that describing them in detail might violate the forum rules. Suffice it to say that the crimes involved murdering many people, including most of his own family (plus he did even more heinous stuff, besides just the murders).

Upon sentencing, the judge says to Angus, "You have been found guilty by a jury of your own peers. I would like to sentence you to life in prison many, many times over, if only I could. Unfortunately that option is not available in this case. You are released without prison time, and without punishment. You may leave the courtroom in the custody of your only surviving relative: your brother Malcolm, who is an upstanding, law-abiding citizen."

Why did the judge not sentence Angus to prison?​

Here's a few clarifications that might help.

• Diplomatic immunity has nothing to do with this riddle.
• The situation happens in a first world country that prides itself on its fair judicial system (the specific country is not relevant). The country (and its legal system) is not corrupt, respects victims' rights and the rights of the innocent, does not torture, does not falsely imprison, does not engage in cruel and unusual punishment, etcetera, etcetera.
• There's nothing broken in the prison system. It's not completely filled with prisoners or anything like that. There are other people still going to prison for other crimes.
• Nobody in the courtroom (including Angus and Malcolm) is facing a terminal illness.
• Angus is not in political office, or in a position of political power.

 

[zoobyshoe]

Spoiler

Would Angus be a conjoined twin, fused to his brother, Malcom?

 

[collinsmark]

Spoiler

Would Angus be a conjoined twin, fused to his brother, Malcom?

Correct! (Golly, I didn't think someone would get the answer so quickly.)

 

[zoobyshoe]

Correct! (Golly, I didn't think someone would get the answer so quickly.)

Spoiler

Mysteries at the Museum has a segment on these guys:

http://en.wikipedia.org/wiki/Chang_and_Eng_Bunker

that I've seen a few times. Once I concentrated on determining the reason imprisonment wasn't an option in this case, this situation soon occurred to me.

 

[Enigman]

Next one:
Rearrange the following letters to make one word:
NEW DOOR

 

[billiards]

Next one:
Rearrange the following letters to make one word:
NEW DOOR

Spoiler

one word

 

[zoobyshoe]

Spoiler

one word

Good one!

 

[Enigman]

Next one:
There are 3 crates in front of you. One crate contains only apples. One crate contains only oranges. The other crate contains both apples and oranges.

And each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples and oranges".

But the labeling machine has gone crazy and is now labeling all boxes incorrectly.

If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you label ALL of the crates correctly?

 

[Enigman]

Another one:

The hour and minute hands are at equal distance from the 6 hour, what time will it be exactly?
---------------
Previous one was too easy. And Sam Loyd was one of the greatest puzzle makers in history...

 

[billiards]

Next one:
There are 3 crates in front of you. One crate contains only apples. One crate contains only oranges. The other crate contains both apples and oranges.

And each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples and oranges".

But the labeling machine has gone crazy and is now labeling all boxes incorrectly.

If you can only take out and look at just one of the pieces of fruit from just one of the crates, how can you label ALL of the crates correctly?

Spoiler

• take label off "apples and oranges box"
• take fruit out of "apples and oranges" box.
• if apple: take label off "apples" box and put on the old "apples and oranges" box.
• take label off the "oranges" box and put on the old "apples" box.
• put the "apples and oranges" label on the old "oranges" box.

if you pick an orange then ammend above instructions by interchanging the word "apples" for "oranges" and vice-versa.

 

[CompuChip]

Good one!

Heh, that reminds me of this old one:

hippomonstrosesquippedaliophobia - with how many letters is it spelled?

 

[Enigman]

Spoiler

• take label off "apples and oranges box"
• take fruit out of "apples and oranges" box.
• if apple: take label off "apples" box and put on the old "apples and oranges" box.
• take label off the "oranges" box and put on the old "apples" box.
• put the "apples and oranges" label on the old "oranges" box.

if you pick an orange then ammend above instructions by interchanging the word "apples" for "oranges" and vice-versa.

Tadashii!
(Correct in japanese)

Heh, that reminds me of this old one:

hippomonstrosesquippedaliophobia - with how many letters is it spelled?

Spoiler

Two.

Another one:

The hour and minute hands are at equal distance from the 6 hour, what time will it be exactly?
---------------
Previous one was too easy. And Sam Loyd was one of the greatest puzzle makers in history...

(Reposted so that no one misses it)

 

[Enigman]

Hint:

Spoiler

Given: time= 8hrs + x min
angle between hands is bisected by 6 hour mark.
1 min--> 1/60 deg of hours hand

 

[Jonathan Scott]

Hint:

Spoiler

Given: time= 8hrs + x min
angle between hands is bisected by 6 hour mark.
1 min--> 1/60 deg of hours hand

Spoiler

Isn't this just trivial algebra? 30 - x = 10 + (x/12) so x = (12*(30-10))/(12+1) = 18 + 6/13

 

[Enigman]

Spoiler

30 - x = 10 + (x/12) so x = (12*(30-10))/(12+1) = 18 + 6/13

Correct though incomplete:

Spoiler

extending the solution: 8hrs 18 min 27 + 9/13 sec.

LaTeX doesn't work in spoilers

 

[Enigman]

An odd bit of history:

Arthur O'Connor of the Society of United Irishmen wanted to petition the French to support an Irish revolution, on his way to France he was arrested and imprisoned in Fort George but on his way to the prison he distributed the following verse:

"The pomps of Courts and pride of kings
I prize above all earthly things;
I love my country, but the king,
Above all men, his praise I sing.
The Royal banners are displayed,
And may success the standard aid.
I fain would banish far from hence.
The 'Rights of Man' and 'Common Sense'
Confusion to his odious reign,
That foe to princes, Thomas Paine.
Defeat and ruin seize the cause.
Of France, its liberties and laws."

-On the first glance it seems he had a change of heart and had become suddenly loyal to the throne but of course, that's not even near the truth...
So what's the true meaning of the verse?

Spoiler

I believe I already gave a hint...did you notice?

Oh and don't google the guy if you don't want to cheat... google him after you get the Enigma and the cat's safely outside the box (or dead).

 

[Jonathan Scott]

Spoiler

I wonder if the verses were written side-by-side?

 

[Enigman]

Spoiler

I wonder if the verses were written side-by-side?

No they weren't...but do go on...box seems to be opening...
Edit-

Spoiler

it seems I oversimplified it a bit by making two different stanzas...edited that...
I will accept your answer as that works too...
Odd-
(1) The pomp of courts, and pride of kings,
(3) I prize above all earthly things;
(5) I love my country, but my king,
(7) Above all men his praise I'll sing.
(9) The royal banners are display'd,
(11) And may success the standard aid:
Even-
(2) I fain would banish far from hence
(4) The Rights of Man and Common Sense.
(6) Destruction to that odious name,
(8) The plague of princes, Thomas Paine,
(10) Defeat and ruin seize the cause
(12) Of France, her liberty, and laws.

 

[Enigman]

Find the smallest palindromic number which is a perfect square and has an even number of digits.

 

[Enigman]

Another one I can't believe I haven't asked yet...
Its a fruity one too!
(though I don't know if Gad's still following the thread...)
In a closed circular room there are 31 people standing such that everyone can see everyone else. Your job is to place a pomegranate such that all except one can see the fruit.

 

[Travis_King]

Another one I can't believe I haven't asked yet...
Its a fruity one too!
(though I don't know if Gad's still following the thread...)
In a closed circular room there are 31 people standing such that everyone can see everyone else. Your job is to place a pomegranate such that all except one can see the fruit.

I'll have a go...

Spoiler

One wall is a mirror. 30 people face the mirror, 1 person faces them with his back to the mirror. Place the pomegranate behind this person.

 

[Enigman]

No mirrors and no smoke...
as its a silly one I will give it to you anyway.

my answer was:

Spoiler

place it on the person's head...

Now the more serious one:

Find the smallest palindromic number which is a perfect square and has an even number of digits.

EDIT: not writing a computer program would be appreciated...

 

[collinsmark]

EDIT: not writing a computer program would be appreciated...

Darn it.

 

[collinsmark]

Find the smallest palindromic number which is a perfect square and has an even number of digits.

Spoiler

Does it have to be palindromic in base ten?

[STRIKE]If not, then I'd say nine (binary: 1001), which is a perfect square of three (3² = 9, or if you prefer in binary, 11 × 11 = 1001), and has an even number of bits (does the term "digit" imply base ten? Hmm).[/STRIKE] Edit: Nevermind, see next post.

 

[collinsmark]

Find the smallest palindromic number which is a perfect square and has an even number of digits.

Spoiler

My new guess is four (ternary 11). It is a perfect square of two, (ternary, 2 × 2 = 11), it is palindromic in base 3, and has an even number of digits in base 3.

 

[Enigman]

Just base ten collinsmark...
(For a smallest number it doesn't seem really small enough...

Spoiler

6 digits

)

 

[Enigman]

Darn it.

Btw collinsmark I did solved it with C++... the clause was just so that I could see if someone's able to solve without programming...I couldn't.

 

[collinsmark]

Btw collinsmark I did solved it with C++... the clause was just so that I could see if someone's able to solve without programming...I couldn't.

Nor could I in base 10. Well, not without incrementally checking numbers manually. Computers are better at stuff like that.

Spoiler

using System;

// This is a C# program that finds the
// smallest perfect square that
// is palindromic in base 10,
// and has an even number of
// digits in base 10.
namespace Enigman_Palindrome_thing
{
    class Program
    {
        static void Main(string[] args)
        {
            int n = 2; // This is the root of the perfect square.
            
            // Now let's loop through the natural numbers to find
            // the palindromic, even digited, perfect square.
            bool foundIt = false;
            while(!foundIt)
            {
                int nSquared = n * n; // this is the number to check.
                string nSquaredString = nSquared.ToString(); // string form.

                // Let the console know which number we're checking.
                Console.WriteLine("Currently checking " + nSquaredString);

                // check if it has even number of digits. 
                if((nSquaredString.Length)%2 == 0)
                {
                    // If we're here it's even. Check if it's a palindrome.
                    bool palindromeFlag = true;
                    for(int i = 0; i<nSquaredString.Length/2; i++)
                    {
                        // If symetric characters are not equal, 
                        // then it's not a palindrome.
                        if (!(nSquaredString[i] 
                            == nSquaredString[nSquaredString.Length - 1 - i]))
                            palindromeFlag = false;
                    }
                    if(palindromeFlag)
                    {
                        // Found it! let's report it to screen
                        // and finish up. 
                        foundIt = true;
                        Console.WriteLine(Environment.NewLine
                            + "Found it! The number is "
                            + nSquaredString + " which is "
                            + n.ToString() + " squared.");
                    }
                }
                // Increment n for next number to check.
                n++;
            }
            // Keep console alive until ready to leave.
            Console.ReadKey();

            // Goodbye!
        }
    }
}

 

[Enigman]

Okay, you probably missed giving the final answer amidst copying your code so:

Spoiler

836^2=698698

 

[Enigman]

There are eight oranges in a box. How can you divide them between eight people so that each person gets one orange, and one orange is still left in the box?
The oranges must not be peeled or cut.

 

[Jonathan Scott]

Okay, you probably missed giving the final answer amidst copying your code so:

Spoiler

836^2=698698

Or more correctly ...

Spoiler

836^2=698896

I see (from a one-line REXX program) that the next one has 12 digits, but I didn't have the patience to find the third one!

 

[Jonathan Scott]

There are eight oranges in a box. How can you divide them between eight people so that each person gets one orange, and one orange is still left in the box?
The oranges must not be peeled or cut.

Spoiler

Is it OK if one person gets a box as well, then?

(Edited to use spoiler tags - I thought I was just asking for clarification of the question!).

 

[Enigman]

Spoiler

Is it OK if one person gets a box as well, then?

Done!
Oh, and spoiler it.

 

[Enigman]

What is the maximum number of kings that can be placed on a chess board so that none of them threatens any other?
EDIT: A better way to put it would be maximum number of independent kings...

 

[Travis_King]

Is that so that they all can't move as moving each King would place itself into check, or that they are positioned in a way that there is a concievable pattern of movements such that all kings could make movements which would ultimately lead to a draw?

 

[Enigman]

Is that so that they all can't move as moving each King would place itself into check, or that they are positioned in a way that there is a concievable pattern of movements such that all kings could make movements which would ultimately lead to a draw?

The only condition is that the largest number of (independent ie. each of a different player or colour) kings should be on board without any king being in 'check' in the arrangement. So, the former follows from the conditions.
See- https://www.physicsforums.com/showpost.php?p=4575363&postcount=754 (same thing with knights)
This one too is a variation of the 8 queen puzzle. (Find an orientation such that there are 8 queens on board without being threatened ('checked') by any other)

A more mathematical restatement would be- in 8x8 square place maximum number of pieces so that there is always more than (or equal to) 1 unit^2 space between two pieces. or something like that...

 

[zoobyshoe]

What is the maximum number of kings that can be placed on a chess board so that none of them threatens any other?
EDIT: A better way to put it would be maximum number of independent kings...

By trial and error only:

Spoiler

16?

 

[Enigman]

By trial and error only:

Spoiler

16?

Correct!

Spoiler

Using logic-
There are 8 rows and 8 columns. On any given row only four kings can be placed, similarly for columns. No. of kings = no. of rows*no. of columns/4=64/4=16
Alternate solution:
http://i.imgur.com/9ZJBt1m.png

EDIT: my solution

Spoiler

http://en.wikipedia.org/wiki/Mathematical_chess_problem first image of the link

 

[Enigman]

(I deleted the previous one posted here on grounds of being too silly)
If it takes four men eight days to dig four holes, how long does it
take one man to dig half a hole?

 

[Jonathan Scott]

(I deleted the previous one posted here on grounds of being too silly)
If it takes four men eight days to dig four holes, how long does it
take one man to dig half a hole?

I find the concept of "half a hole" rather difficult.

Is this meant to be a trivial maths question where the holes are assumed to require a fixed amount of work and "half a hole" means half the amount of work?