checkerboard - How many chess pieces does it take to "cover" all spaces on a chessboard?

Given an 8x8 chessboard, your goal is to "cover" each space on the board with the fewest possible number of pieces. A space is "covered" if there is a piece on it, or if a piece on the board can be moved to that space in one move.

A trivially easy solution would be that a board could be covered with 64 pieces. If you place a piece on every square, every square is obviously covered.

A less trivial solution is 8 - fill an entire row or column with rooks. Obviously, each rook can cover all spaces in its row or column, so the board is covered.

Can this be done with less than 8 pieces? If so, what is the minimum number of pieces required?

Answer

Yes. The minimum number of pieces required is 5.

5 queens can be places such that they cover every space on the board, as in the following example:

There are 12 such arrangements, along with rotation and reflection of each of them.

Edit: The above proves that 5 queens is enough, but it doesn't prove that 4 queens isn't enough. According to this MathOverflow question and its answers, there is no easy logical or mathematical proof, but it has been proven by completely evaluating all possible arrangements of queens on a board. OEIS sequence A075458 gives the minimum number of required queens for any square board from $1\times1$ to $18\times18$.

riddle - I can signal the demise of the world, or I may be the key to unlocking it

I may have four legs, or I may have none-
I may have two ears, but perhaps I cannot hear.
I may have two eyes, but could it be that I cannot see?

Either way, I can be found in your home;
Either way, something trails behind me.
Either way, some may touch me while others refuse.

Maybe I can eat, but maybe I cannot.
Maybe I rely on shadow, but perhaps I rely on light.
Maybe I am scorned, but I can also be loved.

I can signal the demise of the world,
Or I may be the key to unlocking it.
Do you know of whom my tale speaks?

Answer

You might be a

Mouse (small mammal)

but maybe you are a

Mouse (computer accessory)

I may have four legs, or I may have none-
I may have two ears, but perhaps I cannot hear.
I may have two eyes, but could it be that I cannot see?

the first possibility always refers to the mammal, the second to the computer mouse

Either way, I can be found in your home;

OP probably lives somewhere where mouse infestations are more common

Either way, something trails behind me.

tail or cord

Either way, some may touch me while others refuse.

mice (especially the furry ones) scare some people

Maybe I can eat, but maybe I cannot.

same pattern as before

Maybe I rely on shadow, but perhaps I rely on light.

mice hide in the darkness, an optical mouse uses light to detect its position

Maybe I am scorned, but I can also be loved.

mice are considered pests, but sometimes kept as pets

I can signal the demise of the world,

rats and mice seem to have taken cities over in almost every post-apocalyptic movie

Or I may be the key to unlocking it.

https://www.youtube.com/watch?v=vXgXA4VlStE

Do you know of whom my tale speaks?

Not really, no.

A riddle of anagrams

I must get to the bottom of this
But it is so far away
It couldn't be further

So I swap round to be a gardener

I hold up my plants
Holding the nutrients they need

But I change again, my choice
From a range of possibilities
But I am the select

What was I and what do I change into?

Answer

Is it:

Top, pot, opt.
Top is the farthest away from bottom.
Pot hold a plant and its nutrition.
Opt is (thanks to @Will) to make a choice or decision.

pattern - What is an Energetic Transmitter Word™?

_{This is in the spirit of the What is a Word/Phrase™ series started by JLee with a special brand of Phrase™ and Word™ puzzles.}

If a word conforms to a special rule, I call it an Energetic Transmitter Word™.

Use the following examples below to find the rule.

And, if you want to analyze, here is a CSV version:

Energetic Transmitter Words™,Not Energetic Transmitter Words™
AUDIOGRAM,INAUDIBLE
BUCKMINSTERFULLERENE,BUCKY-BALL
COMMUNICABLE,TRANSLATABLE
COMPULSION,OBSESSION
DRAGON,MONSTER
DUMBFOUNDING,BAMBOOZLING

ENTHUSIASM,EAGERNESS
HOMEBUILDER,HOMEMAKER
IMMUNOFLUORESCENCE,EPIFLUORESCENCE 
MICROFAUNA,MACROFAUNA
MOUNTAINEERING,SPELUNKING
TANTALIZES,EXCITES
UNIMAGINATIVENESS,TEDIOUSNESS
IMMUNOTHERAPEUTIC,MICROTRANSPLANTATION

Answer

I think an ETW is one that

contains the letters of a radioactive element (i.e., one with no stable isotope).

Specifics, which also explain why IMMUNOTHERAPEUTIC is extremely energetic:

AUDIOGRAM: RADIUM
BUCKMINSTERFULLERENE: BERKELIUM
COMMUNICABLE: NOBELIUM
COMPULSION: POLONIUM
DRAGON: RADON

DUMBFOUNDING: DUBNIUM
ENTHUSIASM: HASSIUM
HOMEBUILDER: BOHRIUM
IMMUNOFLUORESCENCE: FERMIUM
MICROFAUNA: FRANCIUM
MOUNTAINEERING: ROENTGENIUM
TANTALIZES: ASTATINE
UNIMAGINATIVENESS: EINSTEINIUM
IMMUNOTHERAPEUTIC: CURIUM
IMMUNOTHERAPEUTIC: TECHNETIUM

IMMUNOTHERAPEUTIC: ACTINIUM
(and there are, so Engineer Toast assures me and I have no trouble believing, several others that likewise go into IMMUNOTHERAPEUTIC besides those three)

I confess I haven't checked all the non-ETWs.

riddle - Where did this mysterious note come from?

The boy had a cat on the boat,

Alone among blue all around.
He wrote a very brief note,
Hoping the can would be found.

---

I just came back from the Latpipe Hotel and decided to head to the beach, for a swim. I parked my car and then walked to the sandy shore. After wiping the sweat off my forehead (it was a hot, sunny day), I reached into my bag and pulled out a towel, laying it across a patch of sand. I then knelt down, pulling out my Mohxus swimming gear (that was the name of the brand). But as I stepped on a particular area of the towel, I felt a hard bump from underneath.

I must have not noticed whatever it was underneath the towel, for I soon discovered it was a metal can. I thought it was some kind of seashell or a crab, but it was a can with words strewn across its middle: $\rm Emergency \,Drinking \,Water.$ That was weird... until I opened it and found a rolled up piece of paper inside!

You see, the weird part was not the paper — it was the peculiar writing on it. All I could tell was that it was written in pencil, but I had no idea what it meant!

$<$$\rm u \quad ms,r \quad od \quad \{\rm o \quad\{syr;/$
$\rm O \quad jsbr \quad nrrm \quad om \quad s \quad djo[etrvl/$

$\rm O \quad s, \quad pm \quad s \quad ;ogrnpsy \quad s;pmr \quad eoyj \quad s \quad yohrt/$
$\rm \{;rsdr \quad drmf \quad jr;[/$

What does it mean?

But then I turned the sheet of paper over, and on the back was some kind of... riddle...?

Look for the home of the letters,
Where they are together onboard.
Move East only once without errors,
To welcome the fruit of the sword.

What sword?

And then I noticed something else inscribed on the bottom of the can in very tiny writing. I had to pull out my lucky magnifying glass to see this. (See why it's lucky?)

The elite brAin nEst was born on my name,
But wHen theSp (a king) died, he fell on the same.
This can may hold four-hundred grams,
But now there's two more at the shams.

Are the capitals supposed to be there?

I then looked below and read the words: I had a cat with genus Iztyriu and species Gribvi.

I know nothing about cats. Can you help me?

---

Who was the boy?

What was the cat's name?

Bonus for $50$ rep: What day was it when I went to the beach? (Partial reasoning not accepted.)

---

Hint:

"The pen is mightier than the sword."

Answer

Partial:

The first message is shifted 1 key right on the keyboard, e.g. a->s and s->d. Decoding it gives:

My name is Pi Patel.
I have been in a shipwreck.
I am on alifeboat alone with a tiger.
Please send help.

This references

The movie Life of Pi.

All the clues aren't solved, but I will put this out as a possibility.

The boy is

Pi Patel

The cat is

Richard Parker

riddle - Easy to type, Harder to say

I have two names I often go by —
Name them both to identify
Who I am and where to find me
(This is not hyperbole).

By one name I'm high and mighty,
Often stalked by paparazzi;
Towns, songs, games, films, a war movie
proudly share a name with me.

People see me night and day, and
That's how I want it to stay.
I can be a brilliant way to
Tell someone "Good job! Hurray!"

By the other name some call me.
(Really. Think telephony.)
Or I might be something you'd see
Used during a liturgy.

Truth be told, the name I'm called by
Matters less than how I act —

I might highlight something special
Or from crucial facts distract.
Sometimes giving coders pointers,
Sometimes keeping codes from view,
Sometimes strengthening your strong points,
Sometimes working against you.

I might be close enough to you,
Right now, to spit in your face.
Over 8 times, in this poem,
I appear (check the right place!)

Now it's time for you to answer,
Time to get with the program....
Tell me both the names I go by!
Then show who I really am!

_{Part of a metapuzzles entry: I, for one, welcome our new ...}
 

Answer

You are...

a star/asterisk.

I have two names I often go by —
Name them both to identify
Who I am and where to find me
(This is not hyperbole).

By one name I'm high and mighty,

a STAR

Often stalked by paparazzi;
Towns, songs, games, films, a war movie
proudly share a name with me.

"Star" means "idol", and many movies have "star" in them (including "Star Wars"!).

People see me night and day, and
That's how I want it to stay.
I can be a brilliant way to
Tell someone "Good job! Hurray!"

Stars are in the night sky, and there's one big star in the daytime sky. Gold stars are used for congratulating someone.

By the other name some call me.
(Really. Think telephony.)
Or I might be something you'd see
Used during a liturgy.

"Asterisk" is the term used for the key in telephones, and it represents the Star of Bethlehem.
(from OP:)

Asterisk is a free, open-source IP-PBX telephony system.
And here's a bit more info on the Star of Bethlehem reference.

Truth be told, the name I'm called by
Matters less than how I act —
I might highlight something special
Or from crucial facts distract.

Well yes, highlighting things that are special is certainly common.*
^{*Warning: Only common in websites with Markdown or no text formatting at all.}

Sometimes giving coders pointers,
Sometimes keeping codes from view,

In certain languages, * indicates a pointer type (for example, int* x means pointer to integer). /* ... */ is multiline comment syntax in many C-like languages, and it can be hidden with a press of a button in some editors. (OP adds:)
Asterisks also often replace PINs, credit card/CCV numbers, etc. in entry fields.

Sometimes strengthening your strong points,
Sometimes working against you.

I might be close enough to you,

Right now, to spit in your face.
Over 8 times, in this poem,
I appear (check the right place!)

Well, if you're looking at the post source, you can easily see the asterisks! And the phrase "Over 8" refers to the position of the asterisk on a standard computer keyboard. (OP adds:)
Also, "times", as * is used as a multiplication operator in computing.

Now it's time for you to answer,
Time to get with the program....
Tell me both the names I go by!

Then show who I really am!

You are a STAR/ASTERISK. $*$

logical deduction - Find the pattern of matrices

fill the matrix correctly and the guess the Letter and number of 5th matrix:

PS: I'm sure you won't need options!

Answer

| |X| | | |
|X| | |X| |
| |X| | | |
| | |X| | |
| | |X| | |
  O + 17

The first piece moves to the right, the second moves down, the third moves left, the fourth moves up, and so the fifth probably moves right.

The letters and numbers are the sum of X and Y coordinates respectively, with the upper left corner being coordinate $(1,1)$ or $(A,1)$.

Each image introduce its new element at the position given by the coordinate sum of the previous image.

Both movement and new positions wrap around, so when a piece would move over the border it appears next to the opposite border, similarly a piece that appear at for instance $(H,11)=(8,11)$, will wrap around to $(8\mod5,11\mod5)=(3,1)=(C,1)$.

logical deduction - Numbered Logicians near the Blue Eyes island

There is an island near the famous Blue Eyes island where 100 perfect logicians live, each with a number between 1 and 100 (repetitions allowed) tattooed on their forehead. They know all the numbers of the other inhabitants except their own. If ever one of them learns their own number, tradition dictates that they krill themselves. That is, the next daily all-island meeting, at precisely 12:00 noon, they must dump a bucket of raw krill on their heads.

It so happens that half the inhabitants have the number $50$, while the other half have the number $51$. One day, a travelling knight (truth-teller) is addressing all the inhabitants, and mentions:

I was surprised to find only two distinct numbers among the people here, and that those two numbers were consecutive.

Immediately, all the inhabitants deduce that they have either a $50$ or a $51$ on their forehead. But that is not narrowed down enough to commit "sushi-cide". Do any of the inhabitants end up krilling themselves, and if so when?

Answer

Yes.

One important difference between this problem and the blue-eyes problem is that here, even if there was one 50 and ninety-nine 51s, the person with the 50 would not know their own number (it could also be 52). This means that the crucial chain of deductions cannot start from there.

However, there is a place to start this chain. If there was one 2 and ninety-nine 1s, then the person with the 2 would know their own number. So, if nothing happens after one day, it becomes common knowledge that this is not the distribution.

If there were two 2s and ninety-eight 1s, then both 2s would know their numbers after a day passed, and they would krill themselves on the second day. So if two days pass with no krillings, then it becomes common knowledge that this is not the distribution of numbers either.

After 99 days pass, it becomes common knowledge that the numbers are not 1s and 2s. Then, if there was one 3 and ninety-nine 2s, the person with the 3 would know that they didn't have a 1.

After 198 days, it is common knowledge that the numbers are not 2s and 3s. After the 4851st (99*49) days 49s are ruled out. A similar chain of deductions starting at 100 has also ruled out 52 and higher numbers, so it is common knowledge that the numbers are 50 and 51.

Finally, it takes 49 more days to eliminate all the possibilities except fifty 50s and fifty 51s. On the 4901st day, all of the islanders krill themselves.

mathematics - Coin removal problem

I have a slight variation of the coin flipping problem.

There is a line of $n$ coins on the table; some of them are heads up and the rest are tails up, in no particular order. The object of the puzzle is to remove all the coins by a sequence of moves. On each move, one can remove any head-up coin, after which its neighboring coin or coins, if any, must be turned over. Coins are considered “neighbors” if they are next to each other in the original line; if there is a gap between coins after some moves, the coins are no longer considered neighbors.

Determine the property of the starting line that is necessary and sufficient for the puzzle to have a solution. For those lines that can be removed by the puzzle’s rules, design a method for doing so.

Answer

The row of coins can be fully removed if it has the following property:

It has an odd number of heads

Proof:

By induction.
The property obviously correctly predicts solvability for a row of length 1, where "H" is solvable and "T" is not.
Suppose the property correctly predicts solvability for the row lengths $1$ to $n$.

Case 1: A row of $n+1$ coins with an odd number of heads.
Do a move on the last heads coin, which is at location $m$. This leaves a row of coins to the left of length $m-1$, and a row to the right of length $n-m+1$. If the row to the right is non-empty, it will have one head (at location $m+1$ that you flipped) and the rest tails, so it is solvable by the induction hypothesis. The row to the left will also have an odd number of coins because before the flip it had en even number, so that is also solvable by the induction hypothesis. Therefore the original row of $n+1$ coins is solvable (and a good next move is the last heads of the row).

Case 2: A row of $n+1$ coins with an even number of heads.
Do a move on any heads coin, at location $m$. This leaves a row of coins to the left of length $m-1$, and a row to the right of length $n-m+1$. If either of the two rows is empty ($m=1$ or $m=n+1$), then we have a single row of $n$ coins that still has an even number of heads (one removed, and and coin flipped), so it is unsolvable by the induction hypothesis.
If you really do have two rows now, then one of the rows will have an even number, the other will have an odd number of heads. You can think of the move as flipping three coins in a row (the middle of which was heads) and then removing the middle tails coin to split the row. Flipping the three coins changes the parity of the number of heads to odd, and splitting the row then means that exactly one of the resulting rows must be odd and one even. By the induction hypothesis, one of these rows is not solvable.
The coins become unsolvable whichever move you make, so the original row was unsolvable.

By induction, the property correctly predicts solvability for all values of $n$.

Solving strategy:

The proof shows that you can solve a row by always making your move on the last (or first) heads coin of the row.
Note that making a move in the middle somewhere can sometimes lead to a dead end, where it splits the row into two that are both unsolvable (both have an even number of heads). The simplest example is of course a row of three heads (HHH) which would turn into two separate tails coins if your move was the middle one.

Optimal Building Strategy Puzzle

Suppose a general is in charge of a mining site with 5 workers and he has a total of 50 units of gold. Each worker is able to mine for him 40 units of gold per minute.

The general is able to employ new workers for a total one of cost of 50 gold units (creating a worker takes 17 seconds and each mining site can only construct one worker at a time), but only 24 workers can exist at a mining site. The general can thus decide to construct a new mining site, costing 400 gold units (taking 2 minutes to construct), and with 2 mining sites, he can then construct 2 workers simultaneously, one from each mining site.

The general decides he wants to have a total of 3 mining sites. The general however is unsure of whether he should build any workers to increase income or if he should just wait until he has enough gold to purchase his 2nd or 3rd site.

What is the strategy the general should follow in order to minimize the time taken to achieve the three mining bases he desires? How many villagers should he build from each base and when?

Answer

Should the general build additional bases?

No. A single base is the way to go. The general should just save up to 800 gold units and build both bases at once.

How many workers should the general build?

Five workers, to get a total of 10.

When will the general have completed three bases?

295.5 seconds = 4 minutes and 55.5 seconds. This is two minutes after the first mining site is built at 175.5 seconds.

What build order achieves this time?

Build five workers, starting immediately. One will see that there's enough gold to build a worker as soon as the previous one finishes. So, the limitation is the 17-second build time. These workers are then finished at 17, 34, 51, 68, and 85 seconds.
I used a spreadsheet to compute how much gold one has at each of those times by updating the previous gold with the number of workers and mining rate.
Time Workers Minerals Time to 800 Total time 0 5 50 225 345 Worker built 17 6 56.6667 202.8333 322.8333 Worker built 34 7 74.6667 189.4285714 309.4285 Worker built 51 8 104 181.5 301.5 Worker built 68 9 144.6667 177.2222 297.2222 Worker built 85 10 196.6667 175.5 295.5

After 5 workers, just wait until you have 800 minerals to build both mining sites. The "Time to 800" column says the total time its will take to reach 800 gold if you build no further workers, and 5 additional workers optimizes that time. It allows the bases to be started at 175.5 sec and complete at 295.5 sec.

Why produce that number of workers?

A worker pays for itself after 75 seconds of mining, and so after 92 seconds counting building time. Building a sixth worker would not pay for itself as there are only 90.5 seconds remaining at that point. Building fewer workers doesn't help either, as can be seen from the spreadsheet. At the time the last worker is built, there's more than 92 seconds left before building, so it's worth it.

Why produce that number of mining bases?

Any additional base will have no benefit to use. We are nowhere near the cap of 24 workers on a mining base. Such a base would finish after at least 120 seconds, at which point we've already found it's not worth it to build workers, so additional production won't help. Each base it only worth it to reach to end goal. Since building a base occupies the worker that's building it and slows down mining, it's best to wait until reaching 800 minerals and starting to build both bases at once.

A final batch of four sequence puzzles

This is my seventh (and probably last) batch of sequence puzzles that are nasty and hard to solve; I stress that each of them has a clear and justifiable solution.

---

Sequence 1:
RC, NU, RT, RS, PI, ??, AN, PT

---

Sequence 2:

A, ?, D, ?, G, H, J, K, L

---

Sequence 3:
?, ?, ?, 1, 7, 0, 7, 1, 7, 3, 4, 8, 9, 3, 4, 1, 2, 4, 2, 5, 9, 1, ?, ?, ...

---

Sequence 4:

O, ?, ?, H, R, A, U, ?, U, E, ?, ?, ?, ...

Answer

Sequence 1:

TU

These are the third and fourth letters of each planet in the Solar System.

MERCURY, VENUS, EARTH, MARS, JUPITER, SATURN, URANUS, NEPTUNE
RC, NU, RT, RS, PI, TU, AN, PT

Sequence 2:

S, F

These are the keys on the home row of a standard QWERTY keyboard.

A, S, D, F, G, H, J, K, L

Sequence 3:

6, 0, 7, 7, 6

These are the digits in the decimal expansion of e, shifted down by 1.

e = 2.718281828459045235360287...
digits: 6, 0, 7, 1, 7, 0, 7, 1, 7, 3, 4, 8, 9, 3, 4, 1, 2, 4, 2, 5, 9, 1, 7, 6...

Sequence 4:

U, E, O, H, R, A

These are the second letters of each day of the week, cycled around.

MONDAY, TUESDAY, WEDNESDAY, THURSDAY, FRIDAY, SATURDAY, SUNDAY
O, U, E, H, R, A, U, O, U, E, H, R, A, ...

riddle - What is the name of the board game? #1

^{This is a series of board game riddles, "Name the board game."
Next riddle is here: What is the name of the board game? #2}

From the given poem, name the board game.

We hate cats, we hate cats.
Slip and slide, slip and slide.
Oh the traps, oh the traps.
Rolling aside, rolling aside.

What is the name of the board game?

Answer

I'd say the name of this game is

mouse trap.

because

mice don't like cats and you get trapped and there are balls that roll and slide

visual - Solve my picture riddle! Cowards need not apply!

Something connects these images:

What is the word, and what are the connections?

Answer

They may all be connected by

chicken

for images representing

Chicken Little

Chicken nuggets

Rubber chicken

Chicken scratch

Chicken wire

Chicken drumsticks

And the title 'Cowards need not apply!'

Chicken can mean scared

riddle - The Curse of the Ancients

The night has grown fearful. It has been a fortnight since the rising - when the dead have begun to shamble out of their graves, hunting and preying on sleeping men and women and children. The people are saying the gods are angry; they may very well be right. The land is cursed, and crops are failing. People are starving and terrified.

You have been entrusted by the emperor himself to lead a party to hunt down the source of this curse. Your searches have led you to a giant temple. Aided by two of your most trusted companions, you bravely step inside the giant structure.

"Welcome," an ancient voice speaks. It takes a moment for your eyes to adjust - the room is a hallway, and sitting in front of you is a robed person. "What brings you to this ancient temple?"

"We seek to break this curse on the land," says one of your companions, a soldier in plate mail. He lays his hand on his sword, gripping it tightly. "Our expert led us here. Are you responsible!?"

"...No. But it was our folly that the curse ever came to pass... we were careless. We cannot break the curse, try as we might. Perhaps you may have more luck." the old man says sorrowfully. "But beware! Should you fail, a fate worse than death awaits you. Do you still wish to proceed?"

"Yes," says your other companion, this one a female - a scribe. "We have risked life and limb to get here. We will not balk now."

"Then proceed," the man says, "and may fortune be with you. But beware the traitor! "

You stride into the main chamber - a gigantic spherical room, dark except for the twinkling of small lights all around you, and illuminated panels winding around the wall - you count twenty-four. Inscribed on the panels are symbols - many of which you recognize. In the center of this room is a giant contraption of gears and what looks to be a cylinder suspended in the middle, with a seat on one end. And in front of the contraption are three figures, all hooded.

"Hello," you greet them. They incline their hooded heads and stand up. "We wish to break the curse on the land. How do we proceed?"

"Three names," the middle figure gestures to the contraption. "We ask that you name us."

"Then you will have what you seek," the first man says. This one is clearly a man - his voice is deep and powerful, and he is built strongly. He is clad in leather, and his boots appear rugged. You see something - his hands are red. "Is that blood?" your soldier friend asks.

"...It is. But not my own."

"...So you have killed. Why do you fight?"

"...Fight? I wield my weapon only for myself."

"So you are a mercenary," your friend asserts. "A sellsword who has seen many battles." You hear the distaste in his voice. You recall he hates mercenaries.

"No, that is not my name." the first man said.

"Then why is blood on your hands?"

"It is necessary," he explains. "For survival... and for sport. Not just me, but all humans... well, except for very few."

"War is never necessary," your scribe claims.

"I do not know war," he replies. "I only know how to survive."

You turn to the next person. "Hello."

"Hello," she replies. "You sound nervous. Do not fear - you seem to be a kind and just person."

"Indeed," your scribe says.

"And you as well, young woman. You are well-balanced, too. I like that. Balanced."

"I'm sorry... balanced?" she queries.

"You do not lean too far one way, nor the other," she said. "You have pride, but you are not arrogant. You love, but do not lust. Those are qualities I prize. Balance. ...What do you do for a living?"

She holds up her quill and parchment. "As you can see, I am a scribe."

"..."

"Greetings, friend," you say to the final figure. You hear a ghastly chortle. "I am no friend of yours, human. For eons, I have always been your greatest predator. I claim everything in due time. Fear me."

"Who are you?"

He laughs again, and terror fills you. "You know me. All living things know me. Many try to escape me... but they always fail."

"I do know you."

"Good."

You finally turn to the building itself. The soldier taps your shoulder. "Take a look - scratches and dents in the wall. I see smashed vases, some debris in the distance -" he gestures to the far side of the room, where you see the large panels - "There may have been a battle here... or someone decided to defile the place. What did that old man say? Beware the traitor? Could the traitor have done this?"

"It's possible." You turn your attention to the twenty four panels that line the walls - huge panels, spanning about maybe five hundred feet total, and twenty feet high.

"Twenty-four," your scribe muses. "A good number. It's twice twelve. But there's something else that's strange."

"What is it?" you ask.

"Why are the panels separated so?" she asks. "There are twenty grouped together on the left... but four on the right. They are separated by some distance... perhaps twenty feet? It's bizarre - why ruin such a perfectly symmetric number? This may be important. And the symbols..."

"Those are of the utmost importance," you assert. "Can you copy them down?"

She nods. "Of course. I'll number them as well, from left to right."

You take a deep breath. The fate of the entire world rests on your shoulders.

What do you do?

HINT: This puzzle has references to mythology, but it can be solved without any knowledge of it. Some people have correctly guessed the names of the first and the second people, but none have guessed the third.

To begin, we should first consider the state of the panels on the wall. The question we need to answer is, "Why is there a gap between the sections of panels?". Do that, and you may be able to crack the code. (You may want to look at this question: What characteristics of a ciphertext can be indicators of a particular cipher? ).

open ended - What is the smallest positive number on an 8-segment array? (no repeats)

You are provided with the 8-segment array in the picture together with all the necessary electronics to make any combination of lit segments you want.

Fillable version for graphics. http://i.stack.imgur.com/K1hnN.png

What is the smallest positive number that you can display without any two boxes being identical? (A box is a grey rectangle containing 8 segments)

Any combination of segments is allowed within a box provided it is accepted by the community as a valid symbol. ('accepted' is decided by net up-votes for your answer)

Any recognised numbering system that is commonly used is allowed. If the one you use is common but little-known (perhaps because it is specialized) then you must back up your claim with a link to an authoritative source. You must also explain it precisely so that it can be understood by others.

You cannot mix numbering systems unless you can show that this is routinely done by some substantial section of the populace.

You cannot invent your own symbols or language. All symbols must be understandable and approved by the community.

The competition remains open until there has been no answer for one week. At that point (or as soon after as I am able) I shall accept the most up-voted answer (net votes).

Update

1. 
   

   The question has been put on hold--therefore I'll make an adjustment about the timescale for accepting an answer (if you want to reopen it please vote to do so).

   
   


2. 
   

   If I could wind back the clock, I would require the answers to be finite. This was an oversight because I did not think an infinite could be considered a number.

   
   



3. 
   

   Because there are answers and they have been voted on, it is too late to change the criteria. My suggestion is that people who are interested submit answers in the categories of finite or infinite numbers according to their preference. I will stick to my original criterion for acceptance. Note

   
   

You can display your answer using ASCII provided it is clear how it would look when translated into segments. Otherwise please use an actual 8-segment representation to clarify the layout of any symbols that aren't obvious.

pattern - What is the next shape?

which one is the correct answer of the following picture? What is the algorithm of shapes sequence?

Answer

The answer is A.

Count the number of straight lines on each shape, and it corresponds to the digits of pi. The shapes you provided match 3, 1, 4, 1, 5, 9, 2, 6, 5. The following digit of pi is 3, so the answer is A.

combinatorics - All numbers in a 5x5 Minesweeper grid

Can you place mines on a 5x5 Minesweeper grid such that each number from 0 to 8 appears exactly once?

Good luck!

Answer

Assuming standard Minesweeper rules, here’s one solution (with $ X $ = a mine):

$$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & X & X \\\hline 1 & 4 & X & 8 & X \\ \hline X & 5 & X & X & X \\ \hline X & 6 & X & 7 & X \\ \hline X & X & 3 & X & X \\ \hline \end{array} $$

EDIT: In response to Euphoric in the comments, I solved this purely by logical deduction with a bit of educated guessing to make things easier on me. But if you really want to know how I did it, here’s a rigorous solution:

We’ll start with a blank grid, as such: $$ \begin{array}{|c|c|c|c|c|} \hline & & & & \\ \hline \\ \hline \\ \hline \\ \hline \\ \hline \end{array} $$ Label the rows A-E (uppercase) going from top to bottom, and the columns a-e (lowercase) going from left to right.

The first thing I did was try and place the 0. It cannot be placed anywhere in the central 3x3 square, since that would prevent the 8 from being placed. It also cannot be in any square next to a corner, e.g. Ab, Ad, Be, since that would force the corner it is next to to also be a 0, which is not allowed. The case where it is located in the middle of an edge i.e. Ac, Ce, Ec, Ca requires more work. WLOG, suppose the 0 were placed in Ac. Then, Ab, Bb, Bc, Bd, Ad all have to be safe, which forces Ab and Ad to be 1 and 2 in some order. This, in turn, forces Bc to be 3. Let’s say Ab were 1. Then, there is a mine in one of Aa or Ab. If it were in Ab, then Aa would also have to be 1, so Aa must contain the mine. However, this leads to a contradiction at Ba: it can’t be a mine due to Ab, so it has to be 2 or 3, which are already taken by other squares. (See the grid below. $ S $ = safe.) Contradiction, so the only valid location(s) for the 0 are the corners. $$ \begin{array}{|c|c|c|c|c|} \hline X & 1 & 0 & 2 & X \\ \hline \color{red}{?} & S & 3 & S & X \\ \hline & X & X & X & \\ \hline \\ \hline \\ \hline \end{array} $$
WLOG let’s put the 0 in corner Aa. This makes Ab, Bb, Ba all safe. Looking at their surroundings, we see that Ab and Ba have to be 1 and 2 in some order, so let’s make Ba the 1 and Ab the 2: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & S & X \\ \hline X \\ \hline \\ \hline \\ \hline \end{array} $$ Here, I put Ca as a mine, even though Cb is also another option. Since this is a rigorous write-up, I will explain why Cb cannot be a mine. If it were, then Ca would have to be a 3 and Bb would be a 4: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline 3 & X & X \\ \hline X & X \\ \hline \\ \hline \end{array} $$ By trying out different locations for the 8 (namely, Dc, Dd, Cd, and Bd), we find that none of them allow for all of 5, 6, 7 to be placed. Thus, Cb cannot be a mine.

Returning to our current grid, we need to decide whether Bb is a 3 or a 4. This one’s easier to deduce, as if Bb were a 3, then Cb and Cc would both be safe, and now the 8 cannot be placed anywhere. Thus, Bb is a 4, Cb is safe, and Cc is a mine: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline X & S & X \\ \hline \\ \hline \\ \hline \end{array} $$ Obviously, Cb cannot be 3, so it is either 5 or 6. Here, I made another guess and wrote down Cb as 5, but to be rigorous — if we were to make Cb a 6, then Bd and Dd have to be 8 and 7 in some order, but neither configuration allows 3, 5 to be placed on the grid. Our grid now looks like this: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline X & 5 & X \\ \hline ? & ? & ? \\ \hline \\ \hline \end{array} $$ Only one of Da, Db, Dc is safe, while the other two contain mines. I will show that Da must contain a mine i.e. it cannot be safe. If it were, then it would have to be a 3, which gives us this configuration: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline X & 5 & X \\ \hline 3 & X & X \\ \hline X & \color{red}{?} \\ \hline \end{array} $$ Ea is a mine over Eb since 2 is already taken. However, we can see that Eb is now problematic: it cannot be a mine, but it also cannot be a number as the only valid one it could possibly be is 4, which is already placed in the grid. Therefore, Da must be a mine: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline X & 5 & X \\ \hline X & ? & ? \\ \hline \\ \hline \end{array} $$ Now, there remains one mine between Db and Dc. As it turns out, making either one the mine (and the other the safe square) each give valid solutions, which Marco13 found in their computer search. I chose Dc as the mine for my solution: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline X & 5 & X \\ \hline X & S & X \\ \hline \\ \hline \end{array} $$ Now, Db is either a 6 or a 7. It cannot be a 7, since attempting to place the 8, 6, 3 in the remaining squares is impossible (there will be a leftover square). So, Db is a 6, and the mines must be Ea and Eb, which forces Ec to be a 3: $$ \begin{array}{|c|c|c|c|c|} \hline 0 & 2 & X & & \\ \hline 1 & 4 & X \\ \hline X & 5 & X \\ \hline X & 6 & X \\ \hline X & X & 3 \\ \hline \end{array} $$ From here, it is clear where the 7 and 8 should go (Dd and Bd, respectively), and this gives my final solution.

lateral thinking - Elevator puzzle

A guy lives in the 10th floor of an apartment building. On a normal day he goes to the 5th floor with the elevator and then climbs the remaining 5 floors. On a rainy day he goes to the 10th floor directly.

Why ?

Answer

I think:

He is a short person, so he can't reach the button, when it's raining he has his umbrella to push the button for him