Thursday, 4 April 2013

lateral thinking - How did he know which rose to pick?



One day a wife picked a rose from a magical bush. Little did she know the rose bushed was cursed by a witch and who ever picks a rose from her bush gets turned into a rose and put on the bush. The only way she can get turned back is if her husband picks her as a rose, if he picks the wrong one though he gets turned into a rose also. But here is the thing, the wife has 3 kids and convinced the witch to let her be turned back into a human for the night to see her kids. The next morning he went to the Witch's rose bush and saw 20 roses. After looking at every rose he picks a rose and his wife appears in front of him.



How did he know which rose to pick?



Answer



He picked the rose which




didn't have dew on it.



Because



his wife had been allowed to be a human for the night, but all the real roses had been out overnight and gathered dew.



chess - Can White Castle?


This is the first chess puzzle I composed in the retrograde genre. I originally posted this in a chess dedicated forum. Hope you like it!


In the following position, is it possible that White could still castle?


enter image description here




  • To prove it's possible, all you have to do is provide a legal game.

  • If you believe it's impossible, you need to provide your reasoning.



Answer



Now that we have three increasingly complex proofs (two deleted, one of them mine) that it's impossible, it's pretty clear that it must be



possible after all!



Here's why:




1. b3 Nf6
2. Bb2 Ng4
3. Bf6 gxf6
4. Na3 Ne3
5. Nc4 Nxf1!
6. Ne5 fxe5
7. h4 Rg8
8. h5 Rg6
9. hxg6 Bh6
10. g7 Be3

11. g8=N Bc5
12. Nh6 Ba3
13. Nf5 Ng3
14. Nd4 exd4
15. Qb1 Bc1!
16. Qb2 Nh5
17. Qc3 dxc3
18. Rb1 Nf6
19. Rb2 cxb2
20. Nf3 b1=R

21. Nd4 Ra1
22. Nb5 Ng8
23. Na3 Bb2+
24. Nb1 Bg7
25. e3!! Bf8
26. O-O

In case you haven't already done so, you should totally check out @greenturtle3141's thorough answer (that unfortunately tripped up mere inches before the finish line) to see why the highlighted moves are absolutely essential.
Note that the notation has been edited to work with most PGN viewers, for instance https://chesstempo.com/pgn-viewer.html



This is, without doubt, the most refreshing chess problem I've ever tried to solve. Thanks, OP!


What does it mean for a Rubik's cube to be perfectly scrambled, and how do you reach it? Without doing a checker board



Can you picture the perfectly scrambled cube? A Rubik's Cube that's perfectly scrambled? How would it look? Is it even possible?


The quest for the Perfect Scramble is now upon us, and I challenge you puzzlers to find the state that is "perfectly scrambled" and how to do this scramble, starting with a solved cube, in minimum moves.


But first, before you can even begin to figure it out, mathematically prove it, or even brute force it, you must define it. What exactly is a perfectly scrambled cube? there's 6 faces to the cube, and 9 squares on each one, so how do you define the perfect scramble?


The perfect scramble is the scramble farthest away from the perfect cube. What would that look like?


The quest for the perfect scramble is upon you. It's up to you to solve this Rubik's crisis! Can you do it?



Answer



The "most scrambled cube" is any configuration that requires the greatest number of rotations to solve the cube using a perfect algorithm.


Although unknown, this algorithm is hypothetically called God's algorithm and in fact the maximum number of rotations, called God's number has been found to be 20.


The list of such 20-move cubes is given here.





To define the "most scrambled-looking" cube, consider any legal cube configuration where:



  1. each face displays all six colours

  2. for each face: upon blacking out the cells of any $n$ colours, $n \leq 4$, the resulting $3\times 3$ grid must exhibit no horizontal, vertical, diagonal, or anti-diagonal symmetry

  3. no line of 3 identically-coloured cells may appear horizonally, vertically, diagonally, or anti-diagonally on any face


Alternatively, we might gauge scrambledness via empirical observation as follows:




  1. Given the population, $H$, of all capable human beings on Earth, let $S$ be a subject drawn from $H$, and conduct the following experiment $\forall\; S \in H$:





    1. Explain the nature and objective of the Rubix cube to subject $S$, and provide a "play" cube for familiarization purposes.




    2. After a minimum of 5 minutes of continuous play, present $S$ with a complete set of 43,252,003,274,489,856,000 rubix cubes, one per each legal state of the contraption. For every unique subset of 3 cubes, have $S$ rank the cubes in order of increasing scrambledness per his/her subjective preference.

      After each evaluation, tally $0$ for the cube ranked "least scrambled", $2$ for the cube ranked "most scrambled", and $1$ for the remaining cube.







  2. After all subjects in $H$ have completed all tallies for all subsets of all possible cube states, assign to each cube state a datum $s$ that is the sum of all numbers tallied on the state. $s$ will be a non-negative integer $<$ 1.7$\times$1069.




  3. Let $s_{\rm max}$ be the $s$ datum of greatest magnitude computed in step 3. Let $C$ be the set of all Rubix cube states for which $s = s_{\rm max}$. We conclude that any cube $c \in C$ is "maximally visually scrambled" with 95% certainty.




Wednesday, 3 April 2013

mathematics - 1 2 3 4 5 6 7 8 9 = 100


The sequence of numbers $1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9$ has the property that you can insert mathematical operators in between the numbers from $1$ to $9$ and make the expression evaluate to 100. For example:


$$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 \times 9 = 100$$


There are possibly hundreds of solutions to this problem, involving different varieties of operators. What is the expression with the fewest number of operators inserted (out of the set $+, -, \times, \div$ and maybe $\sqrt{}$ and $!$) that evaluates to 100?




Answer



I believe that this is the smallest:



$123 - 45 - 67 + 89 = 100$



word - Piece de Resistance - Seven Images Are Worth Digits


Seven Images Are Worth Digits


This puzzle is part of the "Piece de Resistance" series. Go back to Part 1 (Ace) for the story.
Ace Two Three Four Five Six Seven ...


Another colourful piece of work, this time on a white background, a bit more stretchy...



enter image description here




Answer



The following is my somewhat crazy interpretation of the rebus, which is either correct or way off:



First image:



The logo for Dash cryptocurrency, which I interpret as "-".



Second image:



A picture of iodine, which I found with a reverse image search that led me to this website which has the same image. I interpret this to be "i" which is iodine's chemical symbol.



Third image:




This shows a chord with three g notes, but only the bottom note is highlighted. I interpret this as low g, or stylistically: "log".



First image inside parentheses:



Dash from the Incredibles movie. I interpret this also as "-".



Second image inside parentheses:



The right part of a bone. I interpret this to be "one".




Putting it all together:



You get $-i\log(-1)$. It turns out that $\log(-1) = i\pi$, so substituting, you get $-i*i\pi$. Now, because $i*i = -1$, the expression simply becomes $\pi$. So I think the answer to the rebus is pi, which the title hints at because it has "digits".



Tuesday, 2 April 2013

nonogram - Autobinomonorownonomicrogram


  (Bonus / bounty follow-up challenges have been moved to Semiïnfinite autobinomonorownonomicrogram)




  ☆   Be the  first  next to make your own nontrivial autobinomonorownonomicrogram   ☆



“Impossible!” you scoff?   Might indeed be, except that your autobinomonorownonomicrogram may be infinitely wide and include leading 0s, as in 01 or 0010. $ \require{begingroup}\begingroup \def \l { \kern-.3em\cdots~ } \def \L { & ~\cdots\kern -.1em } \def \r { ~\cdots } \def \R { \kern-.2em\cdots~\\\hline } \def \p { \phantom{ \Rule {2.5ex}{2.0ex}{0.5ex}} } \def \X {\kern-.5em \Rule{2.5ex}{2.0ex}{0.5ex} \kern-.5em} \def \b {\kern-.5em \p \kern-.5em} \def \1 {\kern-.5em\rlap {\normalsize \bf \kern .2em 1 } \p \kern-.5em} \def \0 {\kern-.5em \rlap{ \scriptsize \kern.3em 0 } \p \kern-.5em} $



Nontrivial  ?” you might ask.   Well, trivial autobinomonorownonomicrograms are just too common.


  $\small\begin{array}{c|c|} \sf\scriptsize Consecutive~counts~(in~binary) \L & 0& 1& 0& 1& 0& 1& 1& 0& 1& 0& 1& 0& 1& \R \l 0 1~~0 1~~0 1~~1 0~~1~~0 1~~0 1 \r \L &\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&\b&\b& \R \end{array}$


is a trivial one that solves to
  $\small\begin{array}{c|c|} \L & 0& 1& 0& 1& 0& 1& 1& 0& 1& 0& 1& 0& 1& \R \l 0 1~~0 1~~0 1~~1 0~~1~~0 1~~0 1 \r \L &\0&\1&\0&\1&\0&\1&\1&\0&\1&\0&\1&\0&\1& \R \end{array}$


Note how the same infinite digit sequence ... 0 1 0 1 0 11 0 1 0 1 0 1... constitutes both the margins’ counts and the interior’s cells. (These counts are binary, so 10 = 2.)   This example is called trivial as...



...nontrivial here means that multiple pairs of adjacent 1s occur among the cells. The example does not qualify because it has only one adjacent pair of 1s.


Autobinomonorownonomicrogram?”   It’s short for auto-bino-monorow-nono[micro]gram.
       auto:   Self-descriptive — cells’ contents match the margin counts’ actual digits.
       bino:   Binary numbers.

     monorow:   Exactly one row tall.
   nonogram:   This type of grid puzzle.
       micro:   No numbers greater than 2, which shows as binary 10 (or 010, 0010, ...).
         (Thus 3 consecutive cells cannot be all 1s.)




Evolutionary path of autobinomonorownonomicrograms. Begin with a familiar nonogram such as this 3×8, where numbers at its left and top margins are length counts of consecutive filled cells in their respective rows and columns.


$$\small\begin{array}{r|c|} & & & 1 & 1 & & 1 & 1 & \\[-1ex] & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 2 \kern.05em \\ \hline 2 ~~ 3 & \b & \b & \b & \b & \b & \b & \b & \b \kern.05em \\ \hline 1 ~~ 1 ~~ 1 & \b & \b & \b & \b & \b & \b & \b & \b \kern.05em \\ \hline 3 ~~ 2 & \b & \b & \b & \b & \b & \b & \b & \b \kern.05em \\ \hline \end {array} \qquad \begin{array}{r|c|} & & & 1 & 1 & & 1 & 1 & \\[-1ex] & 0 & 2 & 1 & 1 & 1 & 1 & 1 & 2 \kern.05em \\ \hline 2 ~~ 3 & \b & \b & \X & \X & \b & \X & \X & \X \kern.05em \\ \hline 1 ~~ 1 ~~ 1 & \b & \X & \b & \b & \X & \b & \b & \X \kern.05em \\ \hline 3 ~~ 2 & \b & \X & \X & \X & \b & \X & \X & \b \kern.05em \\ \hline \end{array}$$


Turn that into binary, where 0s and 1s indicate empty and full cells while binary numbers are used for counts. This already happens to be nontrivial as it has multiple sets of adjacent 1 cells.


$$\small\begin{array}{r|c|} & & & 1 & 1 & & 1 & 1 & \\[-1ex] & 0 & \! 10 \! & 1 & 1 & 1 & 1 & 1 & \! 10 \\ \hline 10 ~~ 11 & \b & \b & \b & \b & \b & \b & \b & \b \\ \hline 1 ~~ 1 ~~ 1 & \b & \b & \b & \b & \b & \b & \b & \b \\ \hline 11 ~~ 10 & \b & \b & \b & \b & \b & \b & \b & \b \\ \hline \end {array} \qquad \begin{array}{r|c|} & & & 1 & 1 & & 1 & 1 & \\[-1ex] & 0 & \! 10 \! & 1 & 1 & 1 & 1 & 1 & \! 10 \\ \hline 10 ~~ 11 & \0 & \0 & \1 & \1 & \0 & \1 & \1 & \1 \\ \hline 1 ~~ 1 ~~ 1 & \0 & \1 & \0 & \0 & \1 & \0 & \0 & \1 \\ \hline 11 ~~ 10 & \0 & \1 & \1 & \1 & \0 & \1 & \1 & \0 \\ \hline \end{array}$$



Empty a couple of corner-cell 1s to attain the micro quality, having counts only of 0, 1 and 10.


$$\small\begin{array}{r|c|} & & & 1 & 1 & & 1 & 1 & \\[-1ex] & 0 &\bf 1 & 1 & 1 & 1 & 1 & 1 &\bf 1 \\ \hline 10~~\bf 10 & \b & \b & \b & \b & \b & \b & \b & \b \\ \hline 1 ~~ 1 ~~ 1 & \b & \b & \b & \b & \b & \b & \b & \b \\ \hline{\bf 10}~~ 10 & \b & \b & \b & \b & \b & \b & \b & \b \\ \hline \end {array} \qquad \begin{array}{r|c|} & & & 1 & 1 & & 1 & 1 & \\[-1ex] & 0 &\bf 1 & 1 & 1 & 1 & 1 & 1 &\bf 1 \\ \hline 10~~\bf 10 & \0 & \0 & \1 & \1 & \0 & \1 & \1 & \0 \\ \hline 1 ~~ 1 ~~ 1 & \0 & \1 & \0 & \0 & \1 & \0 & \0 & \1 \\ \hline{\bf 10}~~ 10 & \0 & \0 & \1 & \1 & \0 & \1 & \1 & \0 \\ \hline \end{array}$$


Almost there, pare down to just one row. The following 1×$\kern1mu\raise1mu\infty$ nonogram would be an autobinomonorownonomicrogram if only its digits were exactly matched. But its cells contain a 1 (circled) where the counts’ digits do not ($\,\scriptsize\wedge\,$).


$$\small\begin{array}{r|c|} \L & 0& 1& 1& 0& 1& 1& 0& 1& 0& 1& \R \l ~0 1~~1 0~~1 \rlap{\kern-.25em\scriptsize\raise-1.5ex\wedge} 0~~1~~0 1 \r \L &\0&\1&\1&\0&\1&\1 \rlap{\kern-.65em\Large\raise-.1ex\bigcirc} &\0&\1&\0&\1& \R \end{array}$$


This nonetheless qualifies as nontrivial because two pairs of adjacent cells contain 1s.


$\endgroup$



Answer



This satisfies the requirements:



enter image description here




I've provided a decimal conversion row and highlighted the locations of 11s in the "clue".


Monday, 1 April 2013

visual - The etched words - Clue Twenty Six


<<---First clue
<---Previous clue





A note from @Mithrandir: I have given explicit permission for @Khale_Kitha to post this, and gave him the answer to encode. If you want to post one, ping @Mithrandir in chat and we'll talk.




The lights dim and you see, behind them, an etching on the wall that you had not noticed previously.


Etched words on a wall


You ponder what the words could mean...




Next clue--->



Answer



I think you are




Calcium Carbonate



A medicine, abbreviated calendar and a cop's criminal with verbal pause;
shortened sugar over retrograte Greek η



η is eta
retrograde makes it "ate", a likely word ending; assume this is charades
over -> on
shortened sugar -> gluc? fruct?

verbal pause is "um" or maybe "uh" or "eh"
cop's criminal -> ?
abbreviated calendar -> cal



Taking the at face value, it looks like



A medicine [def.]; abbreviated calendar [CAL] + a cop's criminal [???] + verbal pause [UM?] + shortened sugar [GLUC?] + over [ON] + retrograde Greek η [ATE<]
which gives CAL???UM GLUC?ONATE



That's suggestive....




trading out GLUC for CARB - this is definition by example, but ok -
and seeing the obvious missing CI in calcium, give us:

CALCIUM CARBONATE   (A medicine [def.]; abbreviated calendar [CAL] + a cop's criminal [CI]
                    + verbal pause [UM] + shortened sugar [CARB] + over [ON]
                    + retrograde Greek η [ATE<])

Wikipedia tells me a confidential or criminal informant (CI) is a thing, so that answers that too. Yay, learning!



Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...