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So... My fellow players and I are in a pickle: we've come across a challenge that is very difficult, and was meant to be impossible. The spoiler has why, but it's not important.
We are level 4, we have a poison based barbarian (eldritch heritage: serptine, animal fury, a dip into monk for grapple stuff), a shuriken based ninja, a hammer and shield warpriest, a now expired cleave fighter (though he'll probably come back with another fighter), and a pseudo-dragon friend.
We're up against 7 level 3 goblins in one room, who when they detect us run to get 2 level 10 (not a typo) hobgoblin guards, and when possible, will involve another level 10 hobgoblin and 2 more goblins. The DM plays them smart too. I don't want to renegotiate the difficulty of the encounter, I want the XP :). The best we can do for positioning is a room with two 5 foot entrances, or a small hallway with 3 entrances. We ran away before, but we're coming back. We've got two weeks to plan this out.
We do have some things going for us: We can milk characters for poison, and could spend a week or so doing such. We also have 5 doses of dragon's bile for poison, which is sure to kill any of the smaller goblins. We've got two bear traps, along with the usual adventuring gear. The place we're attacking is a broken down keep, and the goblins are ordered to not go out for any reason other than killing people.
Aside from leaving and coming back when we're a decent level (there's a somewhat limited time-line here), any ideas on how to "beat" this encounter? Diplomacy *probably* won't work, seeing as we need to kill the goblins for the quest to be complete.
As a side note: Please don't blame the DM for anything, he's new, I've been helping him on request. We could change the encounter, but I personally don't want to do that cause hard stuff is fun.

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You mentioned a character "cheating consistently"- my question is first, how is this character cheating, and which one?
Is this character the reason why the GM is throwing you against some enemies beyond your abilities?
Why even mention this if it isn't pertinent?
Anyway, if you and the other players have willingly allowed this scenario, then it's on you. Either leave and return when better equipped, or try your damnedest to survive.
YOU know that this was to be a nigh impossible challenge- a handful of goblins shouldn't be a problem- but the lvl 10 Hobgoblins should. And if the GM is playing them smart, he should be trying to purposefully kill your party.
The two best solutions is- first: drop the difficulty. You will not survive the encounter against three Hoblins that are double your level.
second: have loot scattered through ruins. somethings that may help you- like goblin bombs and/or alchemist fires, etc...
You could have the Ninja try to kite some goblins into a choke point, where everyone just beats on anything that comes into range.
The GM may be trying to play the goblins smart, but he should realize that Goblins are notoriously fickle and have a non-existent attention span.
Without a "boss" type to tell them what to do, they tend to do whatever they want.

Avoron |
If all of the level 3 goblins are in the same room, they should be pretty easy to deal with. The lack of dedicated spellcasters is a distinct disadvantage, but it's not insurmountable. If you can get your party up to the room undetected, just have the warpriest fill it with obscuring mist and have the martials block the exits. If you're worried about the level 10 hobgoblins hearing the commotion and joining the fight, a silence spell should deal with that problem rather efficiently. The goblins will be completely unable to see, hear, or make noise, and hopefully won't do much more than run around in a panic. Your friendly neighborhood pseudodragon, meanwhile, can use blindsense to find out exactly where everyone is and telepathically relay it to the rest of the party, allowing the ninja to pick off the goblins with poisoned shuriken. Speaking of poison, if you can get your hands on any black spider marsh poison, nightmare vapor, or black keif, you might be able to confuse the goblin horde and get them to fight each other.
The level 10 hobgoblins will obviously be a much more significant challenge, but they should be manageable if you can take them on one at a time, particularly if the barbarian manages to get them into a grapple. It's hard to trump action economy, and if the whole party gangs up on them they can't last for long. I'd also suggest getting some outside help - your best bet is having the warpriest summon an aether elemental or two with summon monster II. These elementals are fantastically overpowered utility summons, capable of flying invisibly up to the hobgoblins and rendering them mostly harmless by flinging their weapons or other essential equipment up to 480 ft away with no save or other defense. Strategically placed bear traps could also come in handy. And don't be afraid to use more obscuring mist or smokesticks to obscure the hobgoblin vision and take advantage of the pseudodragon's blindsense and telepathy; remember, they can't hurt you if they don't know where you are. Probably.

BurningTear |
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Two things:
1st) if someone is cheating make sure he knows how it works, so it's not a mistake, specially if you're all new to the game.
My group is also in it's first campaign and our game barely seems pathfinder because we forget or don't know lots of things, and used to do some pretty OP stuff.
2nd) I really like the silence==>kill all idea, but I'd advise maybe killing them the roleplay way: block the doors and set the building on fire, or block the doors and bring it down on them (stone to mud or something like that existed right?)
The other one I see is going first for the hobgoblins, especially the one that is alone with 2 gobs: cast silence, axe the little ones while the barb grapples the hob, and poke him to death with pointy sticks.
For the other 2, hope the barb can grapple one for long enough that the 3 of you can kill the other one and go help him.
Then the goblins are free xp.
Hope it helps, we also have this problem in that everything is becoming easy because we ignore some handicaps by not knowing them and we've grown a little too strong, I'd like some of those puzzle like chalenges where you can't just kick the door and kill everything in sight.

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The cheating is obvious to everyone: He simply rerolls the dice if he doesn't get what he wants. He's a bit immature, and is probably not a good player to have around, but he's friends with the DM and one other player (not that DM condones his behavior). When he starts doing it, we just make it very obvious that we are watching his dice rolls. The problem is that it causes the DM, a new DM, to over estimate our abilities.
The spellcasting isn't my department, but it definitely seems the way to go. The warpriest seems to be playing more of a fighter than a warpriest, so thinking of spells is typically the last line of thinking.
As stated, I don't want to drop the difficulty, our group is going in again and I can promise that my character won't leave until he's dead or we're done. A death every now and again is good for the soul :P
So, obscuring mist + silence seems good. Summons seem good too. We could potentially block a door to the goblins with rubble, blocking an escape route, but the noise would be risky. Hrmm...

Avoron |
If you'd rather trap the goblins in with physical obstacles than with combat force, using rubble might depend on too many unknown variables. Setting up the bear traps to block the exits sounds fun, but you'd need to attach them to something very quickly and they would only stop the first couple goblins, if that.
A much more effective method would be to have a few martials bring tower shields, using standard actions each round to block the doorways completely.

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Lots of caltrops and maybe tanglefoot bags. Throw the caltrops down ahead of time, then make them chase you. They have to move slow or get injured and slowed anyway. Gives you time to make potshots while also running away.
If the place is flammable, set it on fire.
The tanglefoot bags will slow them down even if they make the will saves.
Anyone have UMD? A scroll of fireball could really help, too.
Also, since you have time to prepare, potions.
Invisibility, heroism, enlarge person. Any buff you'd want.
Maybe even some alchemist fire or acid flasks.
But the caltrops and tanglefoot bags are most important; don't engage in melee until they are hurt and slowed.

GM 1990 |
The issue with the encounter is in the math and probabilities - its statistically unwinnable.
Here's what the unchained "monster creator" template has for a combat CR10 monster averages:
AC:26, t 15, f 19
Saves: +11 +11 +9
CMD: 29
CMB: +13 to +18
HP: 143
Attacks High(Avg Damage):
+18/+13/+8 (28)
Attacks Low (Avg damage)
+13/+8/+3 (21)
Hobgoblin is safely going to hit 2 of 3 attacks doing 56 damage on average -each round-.
Your PCs on the contrary will have to hit AC26 or beat a CMD: 29 to grapple. Even if you grapple it 1 time, its CMB of +18 means it should break the grapple on a roll above 10.
This is just the base mechanics, if your GM actually built it or reskinned a CR10 monster its also going to have things like Damage Resistance, special attacks/abilities, etc.
Its not a matter of desire on your part or if you get some lucky rolls, its in the math - there is no reasonable amount of lucky rolls that would end in success by the party.
Start with addressing the player just not following the rules. Ask the group if you'd let him re-roll if you were playing Monopoly and he didn't like landing on Boardwalk. He's not being a good friend by cheating - I play with an 8, 10, and 12 year old - following the rules is basic societal courtesy, its on the GM to talk to him, but really any of you can raise this to the individual and tell them to quit cheating.
Then discuss with GM about the math. You can send him to the PRD Unchained link for ideas about how hard different CRs end up being. As it discusses, there is a reason APL+3 is recommended high limit for encounters, the game engine has a lot of assumptions about the PCs if you put a CR10 against them...and it is more than 4th level characters will have.

Gevaudan |

If any of the hobgoblins are casters, you guys should not engage. A level 10 caster should have a single spell that wipes your whole party.
If you face more than 1 hobgoblin at a time, you're pretty much dead. Defense barely matters as the hobgoblins are likely around +14 to hit and 60+hp, with solid AC and saves.
Sniping and running makes a lot of sense.
Lots of 1 use items like 10hd fireball necklace orbs make a lot of sense.
Choking them to death with smoke makes a lot of sense.
Luring them into a gauntlet of reflex save traps and quicksand makes a lot of sense.
Hiring two level 3 NPC Master Summoners to nova them with earth elementals probably makes the most sense.

Avoron |
The issue with the encounter is in the math and probabilities - its statistically unwinnable.
No encounter is statistically unwinnable, least of all this one. You're not playing the numbers, you're playing the tactics, and the statistics don't tell the whole story.
First of all, level 10 hobgoblins are CR 9, so all of your numbers should be a bit lower. But that's not what's important. What's important is that they're all situational.
Combat prowess can't help you when you can't see your enemies but they know exactly where you are. A great attack routine won't be so great when your weapon is sitting 480 feet away. A 28 CMD can seem quite low when a whole party is using aid another, and when a ninja is spamming poisoned shuriken a +10 Fort save will start to wear thin.
If any of the hobgoblins are casters, you guys should not engage. A level 10 caster should have a single spell that wipes your whole party.
Yeah, casters are deadly... when they can cast.
Can those hobgoblins cast spells while grappled? Blinded by smoke? Damaged by attacks of opportunity? Sitting in an area of silence? 480 feet away from their holy symbols? Poisoned with some incapacitating condition?
You get the idea. It's not an easy fight by any means, but there are much better options than just giving up because the numbers are too high.

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Yeah, it's probably not feasible for the PCs to win. Maybe if they blew their entire wealth by level into this one encounter; maybe not even then.
An encounter with a CR equal to APL+4 is one where the PCs have ~50% of winning. If the PCs are fully rested and nobody is surprised.
These 3 level 10 hobos are a CR 12 encounter by themselves (assuming they have NPC level wealth). If they have no armor and are wielding clubs, this drops it to a little over a CR 11 encounter.
Both of these are well above your party's APL (4)+4, which equals 8. Hell, ONE level 10 hobgoblin with no wealth wielding a club is a CR 8.
That's a fight that can be won decently well with the ability to set up the battlefield and such beforehand. Three is a really tall (read: maybe impossible) order. And that's before the 9 total goblins.

The Ragi |

Here are some off the wall strategies:
Feather token (Tree) 400 gp
A token that causes a great oak to spring into being (5-foot-diameter trunk, 60-foot height, 40-foot top diameter). This is an instantaneous effect.
- Grab 10 of these, plant all of them on top of the "broken down keep" and watch it collapse under the weight. They can also be used to block passageways - it's up to the GM if they keep growing and knock out the ceiling, or if they stop when they find an obstacle - useful either way.
More expensive alternative:
Necklace of Fireballs (Type I) 1,650 gp
This item appears to be a string or cluster of spherical beads, sometimes with the ends tied together to form a necklace. The spheres are detachable by the wearer (and only by the wearer), who can easily hurl one of them up to 70 feet. When a sphere arrives at the end of its trajectory, it detonates as a fireball spell (Reflex DC 14 half). If the necklace is being worn or carried by a character who fails her saving throw against a magical fire attack, the item must make a saving throw as well (with a save bonus of +7). If the necklace fails to save, all its remaining spheres detonate simultaneously, often with regrettable consequences for the wearer.
- Pool all your money and buy as many of those as possible. Sell all your stuff if you need. Have a character break in the keep, stealthily, find the goblins, and flee while dropping at least a couple of them behind, making sure there’s no hobgoblins around to take them away from the weaker monsters (with lower reflex saves). Try to gather all enemies in one place, and nuke the living crap out of them - have all the party start throwing the rest of the fireballs at the enemies. With any luck, the goblins divided the ones left behind, or even if one hogged them, he'll probably hit someone nearby.
It’s an average 11d6 of damage per necklace, if each player has one, that’s 44d6 of damage in 3 rounds, plus the extra 11d6 damage per necklace that explodes on the targets (if you’re lucky – if you can’t afford more than 4 necklaces, skip the bomb planting part of this plan). Even if the hobgoblins make their reflex saves (who am I kidding, off course they will), there’s still a good chance they’ll croak under the amount of damage.
If they are still alive (barbarian levels with maxed CON?), snipe and run. If they get close enough to full attack, a PC will probably die.

Gevaudan |

You get the idea. It's not an easy fight by any means, but there are much better options than just giving up because the numbers are too high.
So you mean, "If the DM allows them to win by not using any threatening tactics or items or spells or relevant npc skill checks."
I'm pretty sure this is a CR 7 that immediately triggers a CR 11, into a CR 9.5. There are a total of 9 goblins and 3 hobgoblins.
It's fully possible to creatively engage this scenario at 4th level, i.e. flit around the edges and see what you can get done. With lots of escape routes, traps and utility plus creative engagement, the party might make some progress.
Unless the party has 1000's of liquid gp and a magic mart on hand, they will not be able to get the Summon IX scrolls and fireballs necessary to finish this in a conclusive fashion.
Keep in mind, each of those hobgoblins has 10,000gp in gear and stuff.
If anything unexpected happens, the party should expect a quick, brutal wipe.

GM 1990 |
Numbers, statistics, probability, and tactics aside:
As the OP notes, this encounter is being setup because they've had a player cheating dice rolls and the GM wants to challenge them. So lets presume the player keeps doing what they're doing (which is skewing the PCs APL/capabilities), and by great tactics and rolls they defeat this encounter. The GM will just make the next one "level 12s", and the next one level 14s, and so on. An arms-race of GM vs PCs of this kind only ends 1 way - a TPK and unhappy players. So, unless everyone is on board with that kind of game, all the tactics and equipment recommendations is only adding fuel to what is really an out-of-character problem.
If it was my group, which isn't into high mortality rates, I'd rather address the root-cause which isn't the encounter, its a player issue at the table. Just how I'd handle it, the OP knows his group better than the rest of us.

Avoron |
Stop saying stupid s#@& about aether elementals.
Fair enough. I'll let the Bestiary entry speak for me.
blindsense 60 ft
fly 60 ft. (good)
This works as the kineticist utility wild talent telekinetic invisibility, except that it is constant and doesn't end when the elemental attacks. As this ability is inherent, it is not subject to effects such as invisibility purge.
An aether elemental can fling a creature or object (with a maximum range of 480 feet). A creature receives a Fortitude save to avoid being thrown... The aether elemental can throw a creature or object that weighs at most 50 pounds per Hit Die the elemental possesses.

Avoron |
Only for spells that specifically say so, typically with the (object) tag. Telekinetic throw, meanwhile, explicitly states that the saving throw is for creatures.
Anyway, there's no reason to fixate on the elementals; they were just one potential tactic I mentioned. I'm sure the combined power of the internet can come up with even better ones. Gevaudan's suggestion of taking it slow and "flitting around the edges" to accomplish something without a full-scale confrontation also sounds effective.
And sure, maybe it would be a good idea to talk to the GM, at least to let them know that they don't need to keep raising the difficulty of encounters to insane levels. But the OP said they'd rather keep the encounter the way it is and figure out a way to beat it, and I'm trying to help. Why?
cause hard stuff is fun

Bofor |
My GM has on several occasions presented situations that were nigh unwinnable. We as a group have learned to walk away. We have also come to learn that not every situation can be resolved by sword alone. Since spellcasting seems limited, changing the venue to your favor is problematic.
-Trick Hobs into traps/pits
-Change sight environment and be prepared for such change. Smoke/mist
-Liberal use of 'Command' spell from Warpriest to help neutralize Hobs.
-Hit and run/wand of CLW. Try to thin the opposition then run away and heal. Repeat as necessary.
-Accept that TPK is likely. Walk away.

Cevah |

I'm pretty sure attended objects use their attending creature's saves.
Nonmagical, unattended items never make saving throws. They are considered to have failed their saving throws, so they are always fully affected by spells and other attacks that allow saving throws to resist or negate. An item attended by a character (being grasped, touched, or worn) makes saving throws as the character (that is, using the character's saving throw bonus).
The spell can be cast on objects, which receive saving throws only if they are magical or if they are attended (held, worn, grasped, or the like) by a creature resisting the spell, in which case the object uses the creature's saving throw bonus unless its own bonus is greater. This notation does not mean that a spell can be cast only on objects. Some spells of this sort can be cast on creatures or objects. A magic item's saving throw bonuses are each equal to 2 + 1/2 the item's caster level.
/cevah

MajorMishap |
Your Warpriest has Knowledge(Engineering). I'm super new to the game, but can he apply this knowledge to make some basic trip line traps with those alchemical bombs you have? If so, you could rig up one doorway without blocking an avenue of escape by simply collapsing it.
Another simple trap I could think of would be with Spark and some regular lamp oil if you have it. Have your Ninja lure them into the pool of death and have your Warpriest just chuckle as he points a finger and sets them all ablaze!
KILL IT WITH FIRE!
I don't know if I'm helping... or just an idiot that needs to read the Core Rulebook from cover to cover!

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All of the ideas are helping a ton. We meet not this coming Wednesday, but the Wednesday after.
We have some stealthy characters, and so the current sketch is to impose any penalties possible over the next 3-4 days, and then strike again. While potentially *not* the most ethical, we plan on the following:
Attempt to spoil/disease the food and water available to them. This includes several doses of poison in the well, and taking some, diseased material, and spreading it around in their food, but in low enough quantity to prevent actual taste difference (hoping for dysentery).
Traps are hard to use, due to their constant patrolling, so we can't have any too close. The barbarian is fast enough to outrun them, so we'll disturb their sleep often. We won't use any traps until the night we plan on striking, in which the barbarian during his usual sleep deprivation techniques (yelling and banging) will draw any chaser into a trap. We'll decide how far to place the traps based on how far they chase the barbarian (the pseudo-dragon will be tracking that). A couple of spiked pits will probably do *if* we can lure them out.
On the night of the attack, we'll bring in as many summons as possible, but the key is positioning. We believe the outside trap will only work once, and will only get a few, so we'll need some solid ideas on taking them out. Obscuring mist in an enclosed room with the doors sealed shut plus the pseudo-dragon alone could put a decent amount of them to sleep. The rest will probably be resolved in a fair fight. I've yet to talk to the warpriest to see what spells he thinks would aid in this part.
The building is stone, so burning it down will only get the already partially destroyed roof (It's a damaged keep).

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I have a dangerous option, and a better option. The dangerous option will depend a lot on specific details of the ninja player, the keep, the goblins and hobgoblins routines, the GM, and a host of other things that can go wrong.
Dangerous option. Assuming the ninja is somewhat Stealthy. The ninja buys a Tetsubo or other 4x crit weapon, doesn't matter if he is proficient or not, an oil of silence, an elixir of hiding, oil of magic weapon, potion of invisibility, and a potion of darkvision if he doesn't have darkvision. If you have the 3,000 gp for a Flawed Opalescent White Ioun stone he can have proficiency in the Tetsubo. A katana might work, but it is much riskier, only a 2x crit. A spear would be ok with a 3x crit.
The rest of the party causes a distraction to pull out the goblins and hobgoblins. Try to get them all looking one way. The front gate, the North wall, etc. The Ninja sneaks in a different way and finds a spot to hide in for several hours. Once the hobgoblins have gone to sleep, oil of silence, oil of magic weapon, Tetsubo coup de grace. 4d10+2d6 sneak (not sure if you must be proficient)+4+Str+1/2 damage, dc 10+ damage dealt fort save or die. The oil of silence may let the ninja get all 3 hobgoblins if they are asleep at the same time. Drink the potion of invisibility and get out.
If the ninja gets caught, use the potion of invisibility and get out.
Better option. As others have said already. Have the discussion with the cheating player that you don't roll until you get a number you want. The dice giveth, and dice taketh away.

AwesomelyEpic |

I really like the plan of poisoning them and planting traps before the attack, as it will definitely make things easier. I would recommend pellet grenades (they're cheaper than fuse grenades, and different damage types shouldn't make a difference against these enemies) to deal group damage; thunderstones, ghast retch flasks, and flash powder to make them even worse at combat; and caltrops/shard gel and tangleburn bags to slow them down and damage them. As you can probably see, I'm a big fan of alchemical weapons. That should make the goblins insanely easy, and mitigating the threat of the hobgoblins, but even then it'll probably be difficult because they're so high level. If you can somehow purposefully and knowingly get a bag of devouring, then put it on one of the hobgoblin's heads while they're sleeping. It's like a coup de grâce, but even more effective!

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Problem is, shopping for more stuff is out of the question. We're a week away from a decent town, and so we've been using what we have or can make.
We're in a forest, and the dwarf has craft arms and armor, and the ninja had craft poison, with the barbarian and pseudo-dragon milking themselves for a decent supply of poison.