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The symbols of the gems are sapphire, emerald, topaz, and ruby from SW to NE, The colors shining on the carvings should be blue, green, yellow, and red from SW to NE. And yes, you have to pick up a lamp and put it on a different statue to change the color. Just adjusting is not enough.
Cressyk charges the statue left of the crystal door and hits. The adamantine weapon makes a crack in the statue's body.
Round 1
Ghrezzd
Buri
Cressyk
Herman (lamp positioning)
--------
Guardian blue -15HP
Guardian red

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Ghrezzd steps to the 4th statue to the left of the door and takes that latern to the 4th statue to the right of the door; thereby shining the yellow light on the third gem moving SW to NE.
I think that would be two move actions, and at this point Ghrezzd is now holding a latern that should reflect blue light.

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The lanterns all emit white light, the color is reflected by the crystal door, like light through a prisma. And just to inform you, the GM selected four statues which are correct (two left and two right of the door) and it doesn't matter which lantern you put there as they are all the same (white). I hope that makes it easier, it's just a matter of luck ;) And the two statues closest to the door are not there any more, they are now guardians.
Ghrezzd takes a lantern from the left side of the door and exchanges it with the lantern causing the blue reflection. The lamp on the new position also causes the door to reflect blue light on the same carving. Ghrezzd now holds a lantern in his hand.
Round 1
Ghrezzd
Buri
Cressyk
Herman (redo lamp positioning?)
--------
Guardian blue -15HP
Guardian red

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Buri moves up to the second guardian using his totemic powers. The guardians start to attack with their keen scimitars.
Blue vs. Cressyk: 1d20 + 10 ⇒ (20) + 10 = 30 Crit Damage: 2d6 + 10 ⇒ (1, 2) + 10 = 13 + bleed
Blue vs. Cressyk: 1d20 + 5 ⇒ (8) + 5 = 13
Red vs. Buri: 1d20 + 10 ⇒ (8) + 10 = 18
Red vs. Buri: 1d20 + 5 ⇒ (9) + 5 = 14
Crit Confirm vs. Cressyk: 1d20 + 10 ⇒ (20) + 10 = 30
Round 1
Ghrezzd
Buri
Cressyk bleeding
Herman
--------
Guardian blue -15HP
Guardian red
Sorry Cressyk, but at least it's almost minimum damage

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You have Ghrezzd in the wrong spot. I meant 4th from the left... from the perspective of being in the room and looking back at the door. From the previous map, it should have the blue reflecting on the first (and correct) position.
Ghrezzd tries the latern he removed from the statue on the statute to the left. the fifth from the doorway.
"Hold them off! I'll try to get these aligned properly!"
Let me know if you'd rather have me fighting rather than dealing with the laterns.

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Ghrezzd, can you move your token? That would make things much easier ;) I numbered the statues, could you please post your lamp movements for round 1 and 2 again? I reset the positions (or at least I think I did that). And how do you know the correct posstion of the lanterns? The lanterns all emit white light and the color of the reflected light depends on the angle of the light shining at the crystal. Only the GM knows the correct position of the lanterns. A lantern reflected blue on one position may be reflected orange on another position.

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I'm on my iPad. I can edit tonight. I had meant to grab the latern from statue 10 that was reflecting blue, assuming that the latern is creating the color... so I had the blue lantern in my hand. Then I took it to statue 3 that reflecting on the first gem starting in the SE. I assume that would make the blue lantern reflect to the SE gem. Assumming I'm correct, I would have the yellow latern in my hand... but no idea where to put it to reflect on the second gem from the SE. So, my last post was to try the yellow lantern on statue 2.

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Ok, thanks! I think it's a foreign language explaining problem from my side, sorry! I'm not a native speaker as you already noticed. :/ The color is not tied to the lantern, a lantern which shines blue on the gems on one position might shine red on another position. The angle how the light hits the crystal creates the color, not the lantern. Which means if you swap 3 and 10 you have the same result. Lanterns all create WHITE light. Maybe somebody else has more luck with explaining? Just in case it's clear now, you can try again for round 1 and 2.

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What's the bleed damage?
Cressyk notes that his wounds seem to be bleeding more than normal, but continues his assault on the statue, trying to bring it down quickly.
Full-round action: Attack blue statue
Attack (Adamantine Falchion, Power Attack): 1d20 + 8 - 1 ⇒ (17) + 8 - 1 = 24
Damage (Adamantine Falchion, Power Attack): 2d4 + 7 + 2 ⇒ (2, 4) + 7 + 2 = 15
As for the lamps: your best bet might be to get two lamps each and just pick them up and put them down until they match the desired pattern.

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Sorry I was away for a bit. Trying Round 1 again, as Cressyk suggests.
Round 1
Herman picks up two lamps (two move actions, yes?)
Round 2 Herman moves over to statue 7 and tries the first lamp there to see what light it emits when shone on the crystal.
One move action to move, one to try the lamp. Still have second lamp in hand.

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Yes, it's pretty much guessing, but I will give you hints. Bleeding damage is 2.
Ghrezzd picks up a lamp and moves it to the statue to the left. The color of the light changes from yellow to orange. Herman picks up two lamps and puts one lamp to a statue close to the door. The color of the reflected light changes from yellow to red and it shines on the ruby gem engraving. correct Cressyk swings his falchion at the statue in front of him and creates more cracks in the stone.
Round 2
Ghrezzd
Buri
Cressyk bleeding
Herman
--------
Guardian blue -30HP
Guardian red
more hints
You study the angles between the statues and the door. You know what type of light — and where it would shine.
statue 6 shines yellow on gem C
You find another combination of angle and color.
statue 9 shines green on gem B

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Just for my info, you are using your mythic abilities, right?
Buri attacks the guardian with his warhammer and hits, but the hammer does not even chip the smalles piece away. only adamantine can bypass the DR
The statues attack with their scimitars again.
Scimitar blue vs. Cressyk: 1d20 + 10 ⇒ (12) + 10 = 22 Damage: 1d6 + 5 ⇒ (6) + 5 = 11
Scimitar blue vs. Cressyk: 1d20 + 5 ⇒ (2) + 5 = 7
Scimitar red vs. Buri: 1d20 + 10 ⇒ (20) + 10 = 30 Damage: 1d6 + 5 ⇒ (3) + 5 = 8 +bleed
Scimitar red vs. Buri: 1d20 + 5 ⇒ (15) + 5 = 20 Damage: 1d6 + 5 ⇒ (6) + 5 = 11
Confirm Crit 1 Buri: 1d20 + 10 ⇒ (9) + 10 = 19 no
Confirm Crit 2 Buri: 1d20 + 5 ⇒ (4) + 5 = 9 no
Round 3
Ghrezzd
Buri
Cressyk bleeding 2
Herman
--------
Guardian blue -30HP
Guardian red

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I just noticed the "extra standard action" part of the amazing initiative mythic power. Any chance Cressyk could use that a couple of times retroactively? Also, I noticed I was only adding 2 to damage from power attack instead of 3 - his statue should now be at 32 damage, not 30.

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Cressyk's blade slices effortlessly through the air as it hammers the statue again and again.
Standard action (Round 1): Attack blue statue
Attack (Adamantine Falchion, Power Attack): 1d20 + 8 - 1 ⇒ (10) + 8 - 1 = 17
Damage (Adamantine Falchion, Power Attack): 2d4 + 7 + 3 ⇒ (2, 3) + 7 + 3 = 15
Surge (if needed, take the higher roll and add to attack roll): 1d6 + 1 ⇒ (3) + 1 = 41d6 + 1 ⇒ (5) + 1 = 6
Standard action (Round 2): Attack blue statue
Attack (Adamantine Falchion, Power Attack): 1d20 + 8 - 1 ⇒ (4) + 8 - 1 = 11
Damage (Adamantine Falchion, Power Attack): 2d4 + 7 + 3 ⇒ (2, 4) + 7 + 3 = 16
Surge (if needed, take the higher roll and add to attack roll): 1d6 + 1 ⇒ (5) + 1 = 61d6 + 1 ⇒ (3) + 1 = 4

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Round 3
perception: 1d20 + 12 ⇒ (7) + 12 = 19
Know Engineering: 1d20 + 6 ⇒ (4) + 6 = 10
Herman smiles when the first lamp placement sheds Ruby light on the right spot. Moving to the next statue, he sets his other lamp there (on #8). If that doesn't seem right he will move to #9.
Move action to set lantern on #8. Contingent move action to pick it back up if wrong.

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So, Cressyk has spent 3 mythic points - 2 for extra standard actions and 1 for the surge in round 1. He wouldn't have spent the point for the surge in round 2, because he wouldn't have been able to make the to hit roll, regardless.
Cressyk grunts, "Hurry up with those lamps; I can't keep this up much longer." With that, he steps back, retrieves a wand, and taps himself with it twice.
Not an action: 5' step, as indicated
Move action: Retrieve wand from belt pouch
Free action: Mythic point to get an extra standard action
Standard action: Use CLW wand
Cure Light Wounds: 1d8 + 1 ⇒ (5) + 1 = 6
Standard action: Use CLW wand
Cure Light Wounds: 1d8 + 1 ⇒ (2) + 1 = 3

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Herman puts the lantern in his hand on the statue to the right, but the light does not hit the crystal correctly. He moves on and puts the lantern to the next statue in line. This time the crystal reflects green light on the sapphire carving. correct
Cressyk steps back from the statue and starts to heal his own wounds while Buri fights defensively, but does not get through the statue's defense.
Round 3
Ghrezzd
Buri bleeding 2
Cressyk
Herman
--------
Guardian blue -45HP
Guardian red

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Ghrezzd tries his lantern on the first statue. Seeing the light is not reflected correctly he tries the fourth statue and finally the light shines blue on the sapphire gem. correct
The blue statue takes a step towards Cressyk and attacks with his fist.
blue vs. Cressyk: 1d20 + 9 ⇒ (8) + 9 = 17
The red statue swings at Buri.
red Scimitar vs. Buri: 1d20 + 10 ⇒ (1) + 10 = 11
red Scimitar vs. Buri: 1d20 + 5 ⇒ (11) + 5 = 16
Round 4
Ghrezzd
Buri bleeding 2
Cressyk
Herman
--------
Guardian blue -45HP
Guardian red
the remaining lantern is on statue no. 5

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Cressyk puts his wand back in his belt pouch, then beats at the statue with renewed ferocity.
Move action: Stow wand
Standard action: Attack blue statue
Attack (Adamantine Falchion, Power Attack): 1d20 + 8 - 1 ⇒ (13) + 8 - 1 = 20
Damage (Adamantine Falchion, Power Attack): 2d4 + 7 + 3 ⇒ (3, 3) + 7 + 3 = 16
Free action: Get extra standard action, if needed
Standard action: Attack blue statue
Attack (Adamantine Falchion, Power Attack): 1d20 + 8 - 1 ⇒ (7) + 8 - 1 = 14
Damage (Adamantine Falchion, Power Attack): 2d4 + 7 + 3 ⇒ (3, 4) + 7 + 3 = 17
Swift action: Surge
Surge (if needed, take the higher roll and add to attack roll): 1d6 + 1 ⇒ (6) + 1 = 71d6 + 1 ⇒ (1) + 1 = 2

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Buri continues to dance around the guardian ducking and parrying.
#%$! I am not even making a dent on this guy.
Fighting Defensively, AC 24
Attk: 1d20 + 5 - 4 ⇒ (18) + 5 - 4 = 19
Dmg: 1d8 + 2 ⇒ (8) + 2 = 10

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While Ghrezzd determines the position of the last lantern Cressyk and Buri continue to attack the statues. Cressyk hits and finally dispatches the statue, but Buri is not able to connect.
no need to use the surge
Round 4
Ghrezzd
Buri bleeding 2
Cressyk
Herman
--------
Guardian blue defeated
Guardian red

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Assuming Ghrezzd tells the others what he found out
Herman takes the remaining lantern and puts it to the position indicated by Ghrezzd. The light turns to yellow and shines on the remaining gem. Suddenly the guardians lower their weapons, turn around, return to their positions and become statues again. Amenopheus' prison bursts and Amenopheus is free. "I'm sorry for not fully understanding what happened and why, but I recommend to continue on before more of the sanctum’s guardians decide to attack."
The crystalline door is now bisected and can be pushed open easily.
combat over

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Cressyk nods at Amenopheus. "I could use a rest, but if you feel we need to push on.." He proceeds to retrieve his wand and tap himself several more times.
Cure Light Wounds: 1d8 + 1 ⇒ (2) + 1 = 3
Cure Light Wounds: 1d8 + 1 ⇒ (2) + 1 = 3
Cure Light Wounds: 1d8 + 1 ⇒ (5) + 1 = 6
Cure Light Wounds: 1d8 + 1 ⇒ (5) + 1 = 6
Cressyk spent 4 of his 9 mythic points for the day on that combat. Whew.

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"That was some tricky trap. Thanks for taking a beating while we fiddled with it." says Herman clapping Cressyk and Buri on the backs.
The halfling takes a moment to sketch the trap in his journal while the others are healing.
"I am ready to press on whenyou are," says Herman.

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While Cressyk heals up Buri and Amenopheus open the door.
A broad, simple, flat-topped column of sandstone rises from a deep pit in this circular, domed chamber with a white tiled floor. Though no wind can be felt, particles of sand slowly blow around the pit as if circulated by a lazy breeze, their movement creating a soft, rasping susurrus.
Isolated and protected like the treasures and knowledge they sought to preserve, this room served as the Jeweled Sages’ formal meeting hall. The sages had no permanent leader; rather, each member was an equal except when discussing his or her specialty. The specialists would stand on the central pillar to direct debate regarding their chosen subject matter. Upon concluding that business, he or she would again rejoin the other sages. The sand responds to the mental commands of any Jeweled Sage who steps onto it, typically coalescing into a temporary bridge connecting to the column or creating a three-dimensional image to illustrate a point.
The only occupant of this room is a Garundi woman bedecked in the style of an Osirian noblewoman. "Welcome home!" the woman greets Amenopheus with a warm smile. "I'm the Diamond Sage. I was following your endeavors over the past decade, Amenopheus." she admits.
"And why did you decide to remain hidden for so long? How much more might we have accomplished in recent years if we had met sooner?" Amenopheus asks.
"I also want to rebuild the Jeweled Sages, but we disagree on many nuances." she answers.

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Something about the way she said that... Cressyk tries to judge the diamond sage's intentions.
Sense Motive: 1d20 + 9 ⇒ (20) + 9 = 29
Cast detect evil, as well. If any warning bells go off, Cressyk will start casting divine favor, bless, and protection from evil.

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Cressyk thinks that the woman speaks the truth. She wants to work toghether with Amenopheus if they are able to find a consent. The woman is not evil.
You listen to them discussing further.
First, they disagree about the true cause of the Jeweled Sages’ destruction, which evolves into a disagreement about the lost order’s hierarchy. Amenopheus interprets the collapse of the Jeweled Sages as an unfortunate accident driven by outside forces, namely the invasion of Osirion by the Kelish thousands of years ago. Had it not been for this calamity, the Jeweled Sages would have continued to thrive. To Tahonikepsu, the invasion and subsequent purge of the Jeweled Sages reflects a critical weakness in the organization. She posits that, spread out and without a strong leader, the order could not respond quickly and effectively to a new threat. Amenopheus defends the order’s egalitarian organization structure as a key source of strength, allowing each member to act independently when needed. The Diamond Sage retorts that without consistent, decisive leadership, the Jeweled Sages will always be vulnerable and disorganized whenever a new threat strikes.
Second, the two sages disagree on how to rebuild the organization. Both accept that they will need to recover additional sage jewels and recruit new prospective sages. Amenopheus believes that the next generation of sages must be chosen from among the most learned scholars. Not only does this match the founding principles of the organization, but he theorizes that anything less than a well-trained mind would be unable to harness the millennia of memories within each sage jewel. To Tahonikepsu, the thought of populating the Jeweled Sages with cloistered academics might allow for easier retention of historic facts but would undermine the sages’ ability to actually use their knowledge to better the world. The Diamond Sage would rather recruit a generation of active and innovative sages who could act as advisors to powerful leaders, read the current events to predict future decisions, and participate in making history rather than simply recording it.
Finally, both sages have their doubts about each other, no matter how politely they engage in the discourse. From Tahonikepsu’s perspective, despite the Sapphire Sage’s talents, he has lost the trust of Osirion’s ruler. Although his failure is not wholly his fault, the incident may become a liability as the Jeweled Sages attempt to reclaim their place in Osirion. She also believes that Amenopheus’s insistence on embracing a failed model for the Jeweled Sages invites history to repeat itself. On the other hand, Amenopheus is skeptical about Tahonikepsu, for only moments ago she was a complete stranger to him and the Pathfinders. He has served faithfully as the Osirion faction leader for years. Ceding leadership of the restored Jeweled Sages to her seems like an irresponsible gambit, particularly when nobody knows of her qualifications, background, or motives. If presented with such a challenge, Tahonikepsu takes a step back and reverts to her natural, draconic form. "I present my credentials," she retorts. "Now we are strangers no more."
She explains that her kind is dedicated to guarding history and time itself. By combining her longevity, cunning, and patience with the ambitions of a millenniaold society of sages, the Jeweled Sages are certain to thrive.
In summary, Amenopheus wishes to recreate the Jeweled Sages as they were in ages past, whereas Tahonikepsu seeks innovation. Seeing that they have reached an impasse, Amenopheus invokes a Convocation of Sages, an old tradition of the sages to decide the organization’s future with a vote. Each Jeweled Sage receives one vote, though because you represent the Ruby Sage as a group, each of you receives only a fraction of her vote. Understandably, Amenopheus and Tahonikepsu cast their votes in favor of their own policies. You can vote for either Amenopheus’s or Tahonikepsu’s approach to leading the organization, and in doing so you have essentially voted for the future leader of the Osirion faction.
Please vote now :)

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Amenopheus and Tahonikepsu accept your votes. Debating their respective philosophical merits has helped the two sages develop respect for one another, and even though each is certain his or her strategy is the better choice, they both agree that they would be willing to work together in the future and honor Ameonpheus' victory.
Conclusion
With the discovery of another Jeweled Sage and the recovery of a third sage jewel, you have helped Amenopheus recreate the lost Osirian order. Both he and Tahonikepsu agree that the sages shall work with the Pathfinder Society toward their mutual benefit, and they aid you in returning to Eto without further conflict.
The Ruby Sage’s power fades from you as you depart the Pillars of the Sun. Amenopheus is uncertain as to whether or not the ruby sage jewel can be restored, but he begins researching the possibility soon.
The loss of the emerald sage jewel — ostensibly to Grandmaster Torch, a known enemy of the Society — weighs heavily on Amenopheus, and he sends his assurances to his Pathfinder allies that the Jeweled Sages will find Torch and recover their stolen property. What mischief the scarred information broker can cause with such an artifact in hand haunts the Pathfinder Society for months to come.
If you like, you can try to recruit Kafar and Nefti to work for the Society. All I need is a diplomacy check for each of them.