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Saturday, March 25, 2017

double slit experiment - Are photons blinking?

Is a photon passing through a point in space blinking relative to that point?

If the photon for some reason destructively interfere with it self at that point, what happen to it? is it off or is it somewhere else?

By blinking I mean toggling between visible and invisible / detectable and not detectable or maybe fade in / fade out when considering its wave nature.

Edit When the wave goes to zero every half wavelength is it possible that the photon disappears every half wavelength? is it disappearing in way similar to what it does after a destructive interference with another photon only that it will come back completing the wavelength?

Edit
In other words, does the fact that a photon as a particle disappears when the photon as a wave destructively interfere with another, means that the photon as a paricle also disappears at the point in space-time where its wave functon is zero between a crest and a trough? From Anna's answer and comments the photon is not regarded as a wave in its own but i dont understand this.

Answer

The photon is an elementary particle in the standard model of particle physics.

The theory at present has elementary particles as point particles in a quantum mechanical theoretical model using quantum field theory. This leads to the Feynman diagram representation of elementary particle interactions which leads to calculating measurable quantities, as crossections and decays.

The point nature of the particles appears when they are detected, and detection means interaction with other particles or fields. See this singe photon at a time experiment, the photons detected by the point seen on the screen on the left.

single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames

As you see on the left, there are dots, not spread out energy. The wave nature of the photon is in its probability amplitude as expressed by the modulus of its wavefunction, in simple quantum terms. The wave nature can be seen only by an accumulation of photons, which build up the same frequency classical light as seen on the far right.

You ask:

is a photon passing through a point in space blinking relative to that point?

as the first frames on the left show, the answer is no, it acts as a point particle when it interacts.

If the photon for some reason destructively interfere with it self at that point, what happen to it? is it off or is it somewhere else?

Point particles cannot destructively interfere with themselves in the theoretical model of mainstream physics. They leave a footprint consistent with the probability amplitude that describes them mathematically

newtonian mechanics - Have the self sophining beads the same velocity when they reach the ground as the end velocity of freely falling beads?

Look at this:

http://www.youtube.com/watch?v=_dQJBBklpQQ

The chain seems to be moving with a uniform velocity. Is the kinetic energy of all the moving beads in the whole chain (at the time they reach the ground) the same as the kinetic energy (when touching the ground) of the whole chain when you let it fall freely from the same height as the jar? Or move the beads in the chain with the velocity a single bead would have (when reaching the ground) as you let it fall from jar height? In both cases, the chain possesses the same potential energy.

enigmatic puzzle - You have prepared. You are determined. You approach the door

Hints:

You don't need any hints, they are littered throughout the images everywhere, and I'm not giving you any more. :)
Ok, I will perhaps assist in framing things a little... This is actually a deceptively simple puzzle (though, simple $\ne$ easy). It isn't multiple layers deep, even if it may appear that way. In fact, there's really only a single puzzle to be solved here, and I believe it's actually doable with information from only three of the images. Most of the rest are hints and pointers to make your life easier, but also to distract a little. Part of the challenge is in filtering the truly necessary information from the noise, and in taking the lessons learned from the hints and applying them to the actual puzzle.
I'll just add one last bit of extra information (which you shouldn't take as a hint, so much as a clarification, for when you get a certain distance in): The security panel's circuitry contains a paradox prevention system...

Answer

After a period of thought, you approach the door triumphantly in order to type in

2369

as the keycode.

As you do so, you recall the reasoning that led you to this conclusion. Putting together the torn sticker, you saw a number of strange symbols in a gridlike shape. Fortunately for you, a calling card was left on the scene, describing the working mechanism of the locking machine. This struck you as a rather odd security practice, but you were not going to look a gift horse in the mouth.

This indicates that some rays should be traced from some starting points, turning, splitting and stopping according to some rules. You wondered what these rules could be, and reached the conclusion that the strange symbols were surely involved.

The grid had a number of those symbols, but after analyzing all of the available information, the meaning of each slowly became clear.

The easiest one to determine was #, thanks to the help of a previous unworthy puzzler who left his note on the floor. He might have failed in his mission, but you were sure you wouldn't do the same.

The image indicates that a ray incident to the # symbol splits into 3 different rays, one in each direction.

Following that, you remembered another odd piece of paper you had found. > and < didn't take long to fall.

After tracing the paths between the numbers listed, you realize that the paths are perfectly described on the right if the symbols > and < indicate right and left turns, respectively.

The writing on the wall seemed out of place. You thought maybe the placement of the @ symbol was important.

If nothing goes through the @, then you reason it should be a ray blocker of some sort.

The lowercase letters seemed impossible to crack. As a desperate attempt, you look through the peephole, when suddenly it opens to reveal someone looking back at you. Despite nearly dying of a heart attack, you manage to recall some very important schematics inside the room...

You notice that the colors in the drawing are blue, cyan, green, magenta, red and yellow, which have a 1-to-1 correspondence with the lowercase letters in the grid: bcgmry. The bottom picture, coupled with the fact that there are two of each of those letters in the grid, suggested those letters should act as portals, teleporting rays from one point of the grid to the other.

But you were clueless as to how any of this would precisely help you. Even if you figured out the inner workings of the system, how would that possibly help with getting in the door? You go back to the note on the ground...

You thought this could mean there is an input-output relationship between the entered code and what the display shows. Thanks to your fiddling with the terminal earlier, you already knew another input-output pair right off the bat: 0001 = DONTGUESS. You recall an interesting line you saw earlier...

It would make a lot of sense if the display produced the word "ENIGMATIC" when the right sequence was pressed. You now noticed that all the letters necessary to spell ENTERPASSCODE, DONTGUESS and ENIGMATIC were present in the grid, so the words were likely being formed by some interaction between the rays and these letters.

The numbers were still unaccounted for. But with your previous guesses, it only made sense that...

the numbers you typed into the machine were being used as starting points for the raycasting. But there were 4 spaces for numbers, so how could this relate to the points? Would that be related to this part of the note on the ground?

You remember the ramblings of your friend @f'' about combinatorics, and how those numbers represented the number of ways to pick 4 digits with order, 4 distinct digits with order, 4 digits without order and 4 distinct digits without order. You never thought any of that would ever be useful, yet here you are. This clued you to the fact that the order of the digits in the password does not matter: pressing the 0 button causes rays to be shot from all zeros in the grid.

All pieces had nearly fallen into place. But you heard beforehand that most puzzlers that dared to open this door were thwarted by a single symbol. Surely enough, you only had a single symbol left: %, and it was proving to be quite the tricky beast. Maybe...

The previous puzzler's notes suggest % is a line shy from being an X, which indicates blocking of some sort. In its initial state, % is open, but whenever one ray passes through this square, it becomes a blocker for further rays. In conclusion, if two or more rays hit this square, they are stopped on their tracks, but a single ray should be able to make it out.

At this point, all of it was purely conjecture. But remembering you had two output pairs, did they confirm your theories? You had to test it to find out.

What happens when rays from all zeros are cast?
You read the uppercase letters crossed by the rays from top to bottom, left to right: "ENTERPASSCODE". You must be on to something! The next step is to try it out with 0001:
You noticed the pink square is a paradox, because letting a ray through it would allow it to loop around to the pink square, blocking itself. You then assumed whoever engineered the system dealt with this somehow, possibly by blocking the ray outright. Making that assumption, the uppercase letters form "DONTGUESS"! Couldn't be a coincidence, could it?

All that was left was figuring out the combination to open the door. By using some process of elimination to quickly weed out numbers that couldn't be present in the final passcode, you reached your most likely candidate.

Most of the numbers caused the rays to immediately pass through letters not in ENIGMATIC. In fact, the only 4 numbers you saw that did not were 2,3,6 and 9. What happens when you use these 4 as starting points?

You read the crossed letters: ENIGMATIC!

Were you finally worthy? There's only one way to find out. You reach for the keypad...

logical deduction - What are the numbers?

Three perfect mathematicians with extremely strong memories are taking an exam. The examiner tells each of them a certain piece of information about $x$ and $y$, which two positive integers between and including 2 and 100 one is divisible by the other:

The first one is told the difference $d=x-y$

The second one is told the sum $s=x+y$

The third one the ratio $r=x/y$.

Then the following conversation takes place:

First mathematician: I cannot deduce $x$ and $y$ from their difference $d$.

Second mathematician: I knew that you cannot.

First mathematician: Don't bother to say that, I already knew that you knew that I cannot. But I'm not sure if you don't know that I know that you don't !!!!

Third mathematician: What about me...? I did know the two numbers right away, once they told me the ratio $r$!

Question: What are these mysterious numbers $x$ and $y$?

Notes

This is the second puzzle that I created on my own, besides the "consecutive-ranks" puzzle.

I could resume the conversation in just two lines but I preferred to ameliorate it to ease the solution and make it more adapted to logic.

The solution is unique.

The uniqueness is twice verified :D

I will show the solution if this post becomes be sufficiently downvoted or if there are enough unsuccessful attempts

Answer

magic numbers are : (78 , 2 )

First off lets note a_b a couple of two numbers (a*b,b)

Starting from last statement , any set with absent couple such as (2,x) where x > 33 will be excluded

Such couples are:

ratio=35  2-35 
ratio=36  2-36 

ratio=37  2-37 
ratio=38  2-38 
ratio=39  2-39 
ratio=40  2-40 
ratio=41  2-41 
ratio=42  2-42 
ratio=43  2-43 
ratio=44  2-44 
ratio=45  2-45 
ratio=46  2-46 

ratio=47  2-47 
ratio=48  2-48 
ratio=49  2-49  
ratio=50  2-50

So ... standing on that ground , lets begin to work .

this list contains sets of differences that subtractor doesnt know .

set of difference 68 :  2_35 = (70, 35)  4_18 = (72, 18)  17_5 = (85, 5)  

set of difference 70 :  2_36 = (72, 36)  5_15 = (75, 15)  7_11 = (77, 11)  10_8 = (80, 8)  14_6 = (84, 6)  


set of difference 72 :  2_37 = (74, 37)  3_25 = (75, 25)  4_19 = (76, 19)  6_13 = (78, 13)  8_10 = (80, 10)  9_9 = (81, 9)  12_7 = (84, 7)  18_5 = (90, 5)  24_4 = (96, 4)  

set of difference 76 :  2_39 = (78, 39)  4_20 = (80, 20)  19_5 = (95, 5)  

set of difference 78 :  2_40 = (80, 40)  3_27 = (81, 27)  6_14 = (84, 14)  13_7 = (91, 7)  

set of difference 80 :  2_41 = (82, 41)  4_21 = (84, 21)  5_17 = (85, 17)  8_11 = (88, 11)  10_9 = (90, 9)  16_6 = (96, 6)  20_5 = (100, 5)  

set of difference 84 :  2_43 = (86, 43)  3_29 = (87, 29)  4_22 = (88, 22)  6_15 = (90, 15)  7_13 = (91, 13)  12_8 = (96, 8)  14_7 = (98, 7)  


set of difference 88 :  2_45 = (90, 45)  4_23 = (92, 23)  8_12 = (96, 12)  11_9 = (99, 9)  

set of difference 90 :  2_46 = (92, 46)  3_31 = (93, 31)  5_19 = (95, 19)  6_16 = (96, 16)  9_11 = (99, 11)  10_10 = (100, 10)  

set of difference 92 :  2_47 = (94, 47)  4_24 = (96, 24)  

set of difference 96 :  2_49 = (98, 49)  3_33 = (99, 33)  4_25 = (100, 25)

this list contains elements grouped by their similar sum , accepted if summer knows for sure that subtractor doesnt know.

set of sum = 66  :
2_34  -temporarily accepted-  5_13  -temporarily accepted-  7_9  -temporarily accepted-  10_6  -temporarily accepted-  14_4  -temporarily accepted-  

this list is accepted 


set of sum = 68  :
2_35  -temporarily accepted-  3_23  -temporarily accepted-  4_17  -temporarily accepted-  6_11  -temporarily accepted-  8_8  -temporarily accepted-  9_7  -temporarily accepted-  12_5  -temporarily accepted-  18_3  -temporarily accepted-  24_2  -temporarily accepted-  

this list is accepted 



set of sum = 70  :
2_36  -temporarily accepted-  

this list is accepted 


set of sum = 72  :
2_37  -temporarily accepted-  4_18  -temporarily accepted-  19_3  -temporarily accepted-  


this list is accepted 


set of sum = 74  :
2_38  -rejected because 2_38 is unique in the other side of differences- 
 3_25  6_12  13_5  26_2  

this list is rejected 



set of sum = 76  :
2_39  -temporarily accepted-  4_19  -temporarily accepted-  5_15  -temporarily accepted-  8_9  -temporarily accepted-  10_7  -temporarily accepted-  16_4  -temporarily accepted-  20_3  -temporarily accepted-  

this list is accepted 


set of sum = 78  :
2_40  -temporarily accepted-  


this list is accepted 


set of sum = 80  :
2_41  -temporarily accepted-  3_27  -temporarily accepted-  4_20  -temporarily accepted-  6_13  -temporarily accepted-  7_11  -temporarily accepted-  12_6  -temporarily accepted-  14_5  -temporarily accepted-  21_3  -temporarily accepted-  28_2  -temporarily accepted-  

this list is accepted 


set of sum = 82  :

2_42  -rejected because 2_42 is unique in the other side of differences- 


this list is rejected 


set of sum = 84  :
2_43  -temporarily accepted-  4_21  -temporarily accepted-  8_10  -temporarily accepted-  11_7  -temporarily accepted-  22_3  -temporarily accepted-  

this list is accepted 



set of sum = 86  :
2_44  -rejected because 2_44 is unique in the other side of differences- 
 3_29  5_17  6_14  9_9  10_8  15_5  18_4  30_2  

this list is rejected 


set of sum = 88  :

2_45  -temporarily accepted-  4_22  -temporarily accepted-  23_3  -temporarily accepted-  

this list is accepted 


set of sum = 90  :
2_46  -temporarily accepted-  

this list is accepted 



set of sum = 92  :
2_47  -temporarily accepted-  3_31  -temporarily accepted-  4_23  -temporarily accepted-  6_15  -temporarily accepted-  8_11  -temporarily accepted-  12_7  -temporarily accepted-  16_5  -temporarily accepted-  24_3  -temporarily accepted-  32_2  -temporarily accepted-  

this list is accepted 


set of sum = 94  :
2_48  -rejected because 2_48 is unique in the other side of differences- 
 7_13  14_6  


this list is rejected

the previous informations are unnecessary until this point: where subtractor declares his important information:

A list which fits third statement:

set of difference= 68  : 
 2_35  4_18       17_5       rejected because 

 it exists a couple (ab,b)=(85,5) in this set where 2_50 is unique in his set of difference where 
 2_50 has same sum of a_b = 17_5 


 the summer wudnt say i know that you know that i dont 



set of difference= 70  : 
 2_36  5_15  7_11  10_8       rejected because

 it exists a couple (ab,b)=(80,8) in this set where 2_44 is unique in his set of difference where 
 2_44 has same sum of a_b = 10_8 


 the summer wudnt say i know that you know that i dont 



set of difference= 72  : 
 2_37       3_25       rejected because 

 it exists a couple (ab,b)=(75,25) in this set where 2_38 is unique in his set of difference where 
 2_38 has same sum of a_b = 3_25 


 the summer wudnt say i know that you know that i dont 



set of difference= 76  : 
 2_39               4_20               19_5          
 this is the only set accepted because 

 2_39   :  2_39 has same sum of 

 5_15 which has same diff of 
 2_36 which is unique in his set of same sum  

 4_20   :  4_20 has same sum of 
 3_27 which has same diff of 
 2_40 which is unique in his set of same sum  

 19_5        accepted because there dosnt exist any such unique couple  

this set contains the unique solution because there exists two couple  2_39 , 19_5 

 where the subtractor is not sure whether the summer knows if the subtractor knows that summer dosent know the numbers



set of difference= 78  : 
 2_40  3_27  6_14       rejected because 

 it exists a couple (ab,b)=(84,14) in this set where 2_44 is unique in his set of difference where 
 2_44 has same sum of a_b = 6_14 


 the summer wudnt say i know that you know that i dont 



set of difference= 80  : 
 2_41  4_21       5_17       rejected because 

 it exists a couple (ab,b)=(85,17) in this set where 2_44 is unique in his set of difference where 
 2_44 has same sum of a_b = 5_17 


 the summer wudnt say i know that you know that i dont 



set of difference= 84  : 
 2_43       3_29       rejected because 

 it exists a couple (ab,b)=(87,29) in this set where 2_44 is unique in his set of difference where 
 2_44 has same sum of a_b = 3_29 


 the summer wudnt say i know that you know that i dont 



set of difference= 88  : 

 2_45   :  2_45 has same sum of 
     23_3 which has same diff of 
 46_2 which is unique in his set of same sum  


 4_23   :  4_23 has same sum of 
 3_31 which has same diff of 
 2_46 which is unique in his set of same sum  

 8_12   :  8_12 has same sum of 
 13_7 which has same diff of 
 2_40 which is unique in his set of same sum  

 11_9   :  11_9 has same sum of 
 10_10 which has same diff of 

 2_46 which is unique in his set of same sum  


 rejected because all the couples in this set leads to atleast one unique couple in the list of addition  
 which means subtractor knows cetainly that summer doenst know if subtractor knows wheather summer doesnt know numbers 



set of difference= 90  : 
 2_46  3_31  5_19  6_16       rejected because 


 it exists a couple (ab,b)=(96,16) in this set where 2_50 is unique in his set of difference where 
 2_50 has same sum of a_b = 6_16 

 the summer wudnt say i know that you know that i dont 



set of difference= 92  : 


 2_47   :  2_47 has same sum of 
 3_31 which has same diff of 
 2_46 which is unique in his set of same sum  

 4_24   :  4_24 has same sum of 
 5_19 which has same diff of 
 2_46 which is unique in his set of same sum  


 rejected because all the couples in this set leads to atleast one unique couple in the list of addition  

 which means subtractor knows cetainly that summer doenst know if subtractor knows wheather summer doesnt know numbers 



set of difference= 96  : 
 2_49  3_33       rejected because 

 it exists a couple (ab,b)=(99,33) in this set where 2_50 is unique in his set of difference where 
 2_50 has same sum of a_b = 3_33 


 the summer wudnt say i know that you know that i dont

from the last list there s just one accepted set , where there s just one accepted couple specified from last divisor's declaration.

optics - If everything is made up of atoms why doesn't every thing look the same?

At a small scale everything is made up of atoms. Then why is it that objects can have different colours? and why are some objects soft and others hard?

classical electrodynamics - What is the conserved canonical momentum for a relativistically moving charge in a static Coulomb electric field?

The canonical momentum is a fundamental conserved quantity from Noether's theorem for translational invariance of the Lagrangian. Yet I'm finding it very difficult to see its derivation, or even a statement of what it is for something as fundamental as a relativistically moving charge in a static Coulomb electric field.

Can anyone state what it is, or even give a derivation if it's not too much trouble?

Friday, March 24, 2017

grid deduction - WITLESS - A Puzzling Journey

IMPORTANT! READ ME:

The only 'puzzle' parts of this are the links, titled 'Set X', 'Inscription X', or 'Constellation X'. Everything else is flavortext, apart from this readme section, although the flavortext does tell you that the meaning of the inscriptions is not supposed to be deduced until you have finished everything else.

DO NOT even LOOK at the links out of order. They have been placed in this order in order to make the solving experience as interesting as possible.

DO NOT move on from a set until you have solved all 5 puzzles in that set, and have ensured that your interpretation of the rules arrives at a unique solution for all 5.

Every puzzle has a unique solution. If you believe you have found a puzzle which does not have a unique solution, or has no solutions, please comment on this post and I will see if it needs to be fixed.

No puzzle arrives in a solved state. There is always a distinct SOLUTION that must be input into each puzzle.

The mechanics and meanings are one hundred percent consistent, even between sets.

These are plain grid-deduction puzzles, apart from the fact you must work out the mechanics yourself. Once you know the meanings, they are simply pen-paper puzzles in the vein of Japanese puzzles such as Sudoku, Hashi, Nurikabe, Slitherlink, etc.

Feel free to post partials, but SPOILER everything please!

Do not read spoilers unless you have really given up.

The old monk sips quietly from his tea, before placing the cup beside him on the floor.
'So, you wish to climb the Witless Mountain?'
'I do.'
'I am too weak, now, to come with you to the summit. But I can guide you some of the way.' He picks up his staff, and leads the way into the snowstorm, away from the shelter of his shack.
The summit looks so distant - the climb will not be easy.

The snow falls with a certain ferocity, sparkling white pricks appearing on the robe hugged tight against the monk's body.

'Do you see the rockface right up ahead?' His voice is barely audible over the wind.
'Yup!'
'That's your first challenge.'
On the black cliffside, a giant set of shapes and symbols are carved:

Set 1

A few minutes pass.
'I think I understand.'
The monk smiles. 'Very well. Let us continue.'
A second rockface rises around the corner.

Set 2

The monk ushers you along as soon as he sees that the puzzles have been understood.
'It is getting cold. I think this is as far as I can go. I will leave you here.'
'Hang on, aren't you going to tell me what THIS is?'
On the third rockface, a huge row of symbols glows blue beneath the puzzles.
'Ah. You want to know what the blue inscriptions mean?' He laughs wryly. 'You'll be seeing a few of those. I wouldn't worry about them now, if I were you. They're something to think about on the return journey.'
With that, he begins making his way back down the path.
You're on your own, now.

Set 3
(Inscription 3)

Set 4

(Inscription 4)

Set 9
(Inscription 9)

Set 10
(Inscription 10)

Set 11

The peak is so close...

Set 12
(Inscription 12)

The final climb awaits.

Summit

It is dark, now, and as you look up into the sky, the stars begin to shine...

Constellation 1
Constellation 2
Constellation 3

The visions fade, and you return to your senses. You look back down the mountain, past the cliffs, to the gently glowing image of the monk's shack far below. You laugh, recalling his last piece of advice -

And begin the journey down.

EDIT: All 12 sets and constellations have been solved. No inscription solutions have been posted yet - the bounty is for whomever solves the most inscription puzzles!

EDIT 2: Inscription 5 has been taken down for maintenance, sorry to all those who have spent a while on it!

Answer

IMPORTANT! READ ME FIRST!

Try the puzzle before reading this solution. Please, it's worth it.

If you can't figure out a mechanic, take a break and come back. Only look at this guide if you absolutely have to - for example, you've been trying for days and you haven't been sleeping well, or the only way to pacify the three bears currently mauling you is to tell them the solution to this puzzle.

This first section of my answer contains hints for all the sets' "teachings". The first hint is written normally; the second is rot13ed so you don't reveal both at the same time. Only go for the second hint if you've thought about the puzzle with the first for a while and still haven't come to a conclusion.

Set 1

Draw lines on the gray borders. // Fcyvg gur flzobyf rirayl.

Set 2

Last time, you divided the segments into two "equal" sections. // Znxr gur gjb frpgvbaf unir gur fnzr ahzore bs qbgf.

Set 3

You shouldn't need a hint for this one. // Vg'f gur fnzr guvat nf orsber. Rzcgl fdhnerf nera'g "jbegu" nalguvat, gubhtu.

Set 4

You've been making an assumption that you shouldn't have. (I may have reinforced that assumption - sorry!) // Ybbx ng gur frpbaq chmmyr. Vg unf svsgrra qbgf, juvpu lbh znl abgvpr vf abg qvivfvoyr ol gjb.

Set 5

The square mechanic does not interact with the dots. // Ybbx ng gur znexf. Gurl nccrne gb or cbvagvat gb jungrire'f cnfg gur rqtrf.

Set 6

Squares don't need to be separated from each other. // Nqq.

Set 7

It's like a dot, only... different. // Vg'f cnegvnyyl n qbg, naq cnegvnyyl abg n qbg.

Set 8

Again, the symbols are very representative of what's going on. // Ubj qb gur gjb fznyyre gevnatyrf eryngr gb gur ovttre bar?

Set 9

The big triangles can be treated independently. // Gurl whfg fnl gung lbh unir gb or noyr gb nffrzoyr gurz sebz gur unys-gevnatyrf - abguvat zber.

Set 10

Eyes see things. What can these eyes see? // Bar rlr ol vgfrys jbhyq or hfryrff.

Set 11

Come on, really? No, you don't get a small hint for this one. // Ybbx pnershyyl ng gur rlrf gung frrz gb or gur fnzr.

Set 12

These eyes are like the ones from before, only... different. // Vs V chg bar bs gurfr rlrf fvqrjnlf, vg jbhyq abg punatr gur fbyhgvba ng nyy.

I was a budding archaeologist, looking to learn more about the the Witless. The ancient writings had always intrigued me, but I'd never had the chance to make a pilgrimage to the mountain... until now.

As the monk showed me the first rock face, I took out my transparent sheet (so I wouldn't disturb the carvings themselves) and marker, and got to thinking.

That first one was interesting. It appeared to be a domino of some sort, but... that line in the middle was only barely noticeable. Almost as if it was begging to be traced, to be completed.

Yes! That had to be it! But what about the others? Did they work the same way?

They did! I could divide all of them into two "equivalent regions", so to speak.

Ignoring the monk's smile, I moved on to the second rock face. More inscriptions, this time with dots.

So, before I had divided them into "equivalent" regions - could I possibly do the same thing now? The symbols aren't exactly the same, but they could be equivalent in some other way... like the number of dots, maybe?

Bingo.

That wasn't too bad, right? Alright, let's move onto set 3. It seems to be more of the same.

Yep, more of the same. Is this going to get any harder?

...Oh.

The First Roadblock (Set 4)

There's definitely no way to split that into groups of 6.

But wait...

who said it had to be two groups of six?

There we go. Does it work for the rest, too?

Alright, it does! So maybe I was wrong about

the rule being to split into two equal groups - it seems to be "split into any number of equal groups". Cool, so far, so good.

The Second Roadblock (Set 5)

Okay, what.

What is this.

This is not a dot. I liked dots. Dots were sensible, and logical, and they taught me things. But the square is not helping right now.

So what could it be? Maybe...

a negative dot?

Nope, that fails at the fourth carving.

What about...

being "worth" the area of that region?

Nope, that fails at the fourth carving too.

Wait, maybe I was wrong in both of those ideas.

Maybe it's not worth anything at all! Maybe it tells us something else.

But if so, then what is it?

It looks like it should be something related to the borders... specifically, the things outside the borders...

Aha!

It's the number of regions that that region borders!

And that works for all of them!

Let's continue to the next set - it works the exact same way.

Semidots

Okay, this symbol makes sense. It's like a dot, only... not a dot. I guess that means that...

the dots are optional?

Yep, seems right to me. Moving on.

Triangles

Ooh, new symbols. Okay, so we have two small triangles and one big one.

It seems like the two triangles fit together to make the big one. Maybe that logic carries over somehow to the regions themselves?

Let's see if that works:

It does! And it also tells us

that we can't rotate the half-triangle pieces, since the last two puzzles would have multiple solutions if we could.

Oh, this time there are multiple big triangles!

I guess it works the same - each big triangle gets its own copy of the two half-triangle region shapes. They can be combined in different ways, too, it looks like.

The Third Roadblock

Well, that's... interesting. Now we have eyes.

Maybe I should turn around before I accidentally activate some sort of ancient curse.

No, no, I came here to get to the top, and that's exactly what I'm going to do.

^{_{I just don't want to look at the eyes too much.}}

Anyway, eyes look at things. (Or so I've heard.) Maybe that's somehow related to what they do?

Maybe what they "see" is important?

But how?

It can't be what they see in their own region. The third puzzle rules that out, since the eye in the middle can't see everything in its own region. (We can solve that one without even knowing what the eye does, which is nice.)

Maybe it's that...

they all have to see the same thing?

Yep, that seems to work. And it gives more unique solutions!

This also teaches us that we can't have any "dangling" walls - all walls have to be between two different regions.

So, the next set... it seems to have the eyes again, but rotated and reflected.

That just means what they "see" needs to be rotated and reflected too! This isn't so bad.

Alright, one set to go! And it has the swirly eyes. I guess those just mean...

that we don't have the direction. (But hey, at least the reflection is still given!)

Wait a minute. I was a fool.

How could I think those original puzzles were so simple? Of course the symbols meant something. No, I have to go back there with my newfound knowledge and do it right this time.

The Summit (Partial)

It's cold up here.

I have no idea what I'm doing.

(Okay, I have a vague idea of what I'm doing, but it doesn't seem to be helping much.)