mathematics - IX-NAY on the IX-SAY

Will this sequence ever have a 6 in it?

9, 1, 1, 1, 10, 3, 1, 1, 10, 5, 1, 1, 10, 1, 5, 2, 1, 1, 10, 1, 1, 1, 5, 4, 1, 1, 10, 3, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10, 5, 1, 1, 5, 3, 1, 1, 5, 4, 1, ...

Daily hint #1:

The title gives away a lot, but as always, we are looking for a proof, which turns out to be surprisingly elusive. In fact, I just found a hole in my own proof, too. Oops. Well, at least the answer is still the same :-)

Daily hint #2:

@phenomist has already correctly identified the sequence, and the role of the first number in it.

It turns out that if you choose a different starting number, you get a different sequence, and sometimes, a different answer altogether! For example, here's what happens if you start with a 14:

14, 1, 10, 1, 1, 1, 5, 1, 1, 1, 10, 3, 1, 1, 5, 3, 1, 1, 10, 5, 1, 1, 5, 5, 1, 1, 10, 1, 5, 2, 1, 2, 5, 2, 1, 1, 10, 1, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 10, 3, 1, 2, 5, 2, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10, 6, ...

Daily hint #3:

I was planning to put my code for generating these sequences on tio.run for today's hint, but alas, I've used a CPAN module that's not available there. In case you have perl5 installed, and have the means to install the missing module (Math::Roman), here's the code on pastebin.

Also, there's another hint embedded in the code itself: it outputs a newline after every occurrence of a "10". That might prove helpful :-)

Final hint: (edited to make it even hintier; the bounty is running out of time)

Here's the start of the sequence with a lot of extra formatting:

 *: 9, 
 A: 1, 1, 1, 10,
 B: 3, 1, 1, 10,
 C: 5, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 10,
 A: 1, 1, 1, 5, 4, 1, 1, 10,
 B: 3, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 C: 5, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 A: 1, 1, 1, 5, 4, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,

 B: 3, 1, 1, 5, 1, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 C: 5, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 A: 1, 1, 1, 5, 4, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 B: 3, 1, 1, 5, 1, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10,
 C: 5, 1, 1, 5, 3, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 5, 1, 1, 1, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 1, 10,
 D: 1, 5, 2, 1, 1, 5, 5, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 5, 1, 2, 5, 2, 1, 2, 5, 2, 1, 1, 5, 3, 1, 1, 5, 4, 1, 2, 5, 3, 1, 1, 5, 4, 1, 1, 5, 3, 1, 1, 5, 5, 1, 1, 5, 4, 1, 1, 5, 5, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10
 

steganography - Find the number using the data

There is a number hidden in this picture

The number is not 365

This is all I can tell you for now

Answer

The number is:

24 (Original guess: 1)

Explanation:

The math expression of the picture (could be seen in edit mode) is 36*8/12, making 24.

(Original guess: "First blank page" and the simple instruction: "Write a good one".)

knowledge - Cryptic Rebuses

A cryptic rebus, so says I, is a rebus that has the form of a cryptic clue. That is to say, there are two parts to a cryptic rebus:

• a definition of the answer


• a subsidiary indicator of the answer.

A definition can take the form of a picture which straightforwardly represents the answer, or it can be a synonym of the answer.

The subsidiary indicator of the answer, on the other hand, will be some kind of Rebus word/image play which leads to the answer.

Here is an example:

This clue depicts a donkey beside a ‘D’ on a key. The image of the donkey constitutes the definition of the clue, and the ‘D’ ON KEY Rebus constitutes the subsidiary indicator.

This simple example illustrates the structure of a cryptic rebus.

As with purely verbal cryptic clues, however, it will not always be clear what constitutes the definition and what constitutes the subsidiary indicator. Unlike verbal cryptic clues, where the definition must occur at the beginning or end of the clue, the definition of a cryptic rebus can appear anywhere. It can even appear misleadingly integrated in or around the subsidiary indicator. (Perhaps it can even lend positional relations to the subsidiary indicator!) Part of the fun of these puzzles is the thrill the solver gets upon discovering where exactly to “split” the clue into its components.

Also as with purely verbal cryptic clues, there is an exception to the rule that all clues consist of a definition and a subsidiary indicator. So-called double (or triple) rebuses comprise multiple rebuses, each one of which might yield a definition which points to the answer (as in standard cryptic double definitions); alternatively, one (or two) might yield a definition while the other directly yields the answer.

Here's your first set:

_{*I don't own any of the images from which I drew in making these puzzles. Where possible I tried to use public domain images. I believe I may be using the rest in accordance with fair use.}

Answer

Partial answer

3.

BOLD MOVE = "courageous action"

4.

FALL DOWN = "trip"

7.

FRENCH KISS = "swap spit" = "bisou" ("bisou" is the French word for "kiss")

10.

MINTING (M IN T IN G) = "making cents?" ('?' indicates definition by example)

13.

CORRUPTED (C OR R (UP TED)) = "spoilt"

15.

EISENSTEIN (sounds like "'i's in stein") = "Russian director" (the Cyrillic is a transcription of "director")

geometry - Tiling rectangles with N pentomino plus rectangles

Inspired by Polyomino Z pentomino and rectangle packing into rectangle

Also in this series: Tiling rectangles with F pentomino plus rectangles

Tiling rectangles with T pentomino plus rectangles

Tiling rectangles with U pentomino plus rectangles

Tiling rectangles with V pentomino plus rectangles

Tiling rectangles with W pentomino plus rectangles

Tiling rectangles with X pentomino plus rectangles

The goal is to tile rectangles as small as possible with the N pentomino. Of course this is impossible, so we allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one N-pentomino will tile. Example shown, with the $1\times 1$, you can tile a $2\times 4$ as follows:

Now we don't need to consider $1\times 1$ any longer as we have found the smallest rectangle tilable with copies of N plus copies of $1\times 1$.

There are at least 22 more solutions. More expected. I tagged it 'computer-puzzle' but you can certainly work some of these out by hand. The larger ones might be a bit challenging.

mathematics - 5 Pirate Puzzle Question

Question: 5 pirates of different ages have a treasure of 100 gold coins.

On their ship, they decide to split the coins using this scheme:

The oldest pirate proposes how to share the coins, and ALL pirates (including the oldest) vote for or against it.

If 50% or more of the pirates vote for it, then the coins will be shared that way. Otherwise, the pirate proposing the scheme will be thrown overboard, and the process is repeated with the pirates that remain.

As pirates tend to be a bloodthirsty bunch, if a pirate would get the same number of coins if he voted for or against a proposal, he will vote against so that the pirate who proposed the plan will be thrown overboard.

Assuming that all 5 pirates are intelligent, rational, greedy, and do not wish to die, (and are rather good at math for pirates) what will happen?

Here's the solution: The oldest pirate will propose a 98 : 0 : 1 : 0 : 1 split, in other words the oldest pirate gets 98 coins, the middle pirate gets 1 coin and the youngest gets 1 coin.

Let us name the pirates (from oldest to youngest): Alex, Billy, Colin, Duncan and Eddie.

Working backwards:

2 Pirates: Duncan splits the coins 100 : 0 (giving himself all the gold). His vote (50%) is enough to ensure the deal.

3 Pirates: Colin splits the coins 99 : 0 : 1. Eddie will accept this deal (getting just 1 coin), because he knows that if he rejects the deal there will be only two pirates left, and he gets nothing.

4 Pirates: Billy splits the coins 99 : 0 : 1 : 0. By the same reasoning as before, Duncan will support this deal. Billy would not waste a spare coin on Colin, because Colin knows that if he rejects the proposal, he will pocket 99 coins once Billy is thrown overboard. Billy would also not give a coin to Eddie, because Eddie knows that if he rejects the proposal, he will receive a coin from Colin in the next round anyway.

5 Pirates: Alex splits the coins 98 : 0 : 1 : 0 : 1. By offering a gold coin to Colin (who would otherwise get nothing) he is assured of a deal.

(Note: In the final deal Alex would not give a coin to Billy, who knows he can pocket 99 coins if he votes against Alex's proposal and Alex goes overboard. Likewise, Alex would not give a coin to Duncan, because Duncan knows that if he votes against the proposal, Alex will be voted overboard and Billy will propose to offer Duncan the same single coin as Alex. All else equal, Duncan would rather see Alex go overboard and collect his one coin from Billy.)

I don't really understand the 4 pirates case. Why wouldn't they choose 99:1:0:0 instead? Isn't this also going to be accepted?

Answer

Colin would not vote for a 99 : 1 : 0 : 0 split, because if this gets rejected, then Colin would become the leader and get 99 coins, which is better than 1 coin. Duncan and Eddie obviously will not vote for this either since they would get nothing if it was accepted. So this split would not get accepted, meaning Billy would not propose it.

mathematics - Strategy to beat the Casino

Two players, $A$ and $B$, and the Casino play a game. It involves showing zeros or ones: each player picks $0/1$ and also the Casino picks $0/1$.

• If the Casino and both players $A$ and $B$ show the same number ($000$ or $111$), then the two players win the game.


• But in case not all three chosen numbers are identical, the Casino wins.

Altogether there are nine rounds. Now it happens that player $A$ is going to learn some illegal information, just seconds before the game starts: $A$ is going to learn the Casino's choice for each of the coming nine rounds. Unfortunately, there is no way of communicating this information to player $B$ without the Casino noticing. The only way of communicating is via the cards chosen by $A$.

The evening before this game, $A$ and $B$ meet and agree on a common strategy. Is there a strategy that guarantees them to win at least $5$ of the $9$ rounds? And is there a strategy that guarantees at least $6$ wins?

riddle - Alive, dead, alive, now dead

I communicated with you, not by normal means.
The illusionist did everything it seems.
Proximity can make me very shy.
Without a bunch of these you would die.
The bee, the seasons, maybe some more,
Who am I? All alone, no knock on the door.

Hint 1

Who is important.

Hint 2

Don't take the lines too literally (except the one about dying).

Hint 3

The title is important too.

Hint 4 (this is a big one).

Think of each line separately, and then answer the riddle.

Hint 5

This has a similar format.  

Answer

You are

The Police, who have disbanded, reunited, then disbanded again.

I communicated with you, not by normal means.

"Message in a Bottle"

The illusionist did everything it seems.

"Every Little Thing She Does Is Magic

Proximity can make me very shy.

"Don't Stand So Close to Me"

Without a bunch of these you would die.

"Every Breath You Take"

The bee, the seasons, maybe some more,

Band members Sting, Andy Summers, and others

Who am I?

The Police!

All alone, no knock on the door.

This I am unsure about. They have other song titles that sort of fit ("So Lonely", "Hole In My Life"). Maybe there's a particular lyric that fits better?