EDIT:

I'm updating the riddle to what the authors should have specified so that people who see this post and try to solve it for fun don't get frustrated:

You are in a jail cell, surrounded by gold coins. You offer the guard at the door a gold coin and he opens the door. He tells you that there are are 10 guards in total, him included, and they will let you pass them once for one gold coin. Each time you pass one you must pay the toll, even if you're going back. You are not allowed to pick up more than 4 coins at any time, meaning when passing a door you will be left with a maximum of 3 after the bribe, but you may store any number of coins in front of any door. What is the minimum number of coins that must be in the first cell in order to bribe all the guards and escape?

Original post below:

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Reposting here, r/riddles pointed me here. Alright, so it goes like this, it's been killing me:

You are in a jail cell, surrounded by gold coins. You offer the guard at the door a gold coin and he opens the door. He tells you that there are 9 guards in total and they will let you pass them once for one gold coin. Each time you pass one you must pay the toll, even if you're going back. You are not allowed to take more than 4 coins with you, but you may store coins in front of any door. What is the minimum number of coins that must be in the cell in order to bribe all the guards and escape?

I've tried doing the math starting the last door moving towards the first, but apparently I'm not correct, solution I've used below, please share any suggestions if you have them.

cost of resupply= number of resupply runs *2 (one coin to go back, one to pass the guard again)

resupply=how many gold coins you need to bring from the previous door storage to your current one

escape:0 total : 1 consumed to get here

9:1 total : 1 consumed to get here

8:2 total : 1 consumed to get here

7:3 total : 1 consumed to get here

6:4 total : 1 consumed to get here

5:8 total ; 5 stored -- 3 left at door, 1 consumed to get here, 2 cost of resupply, 2 resupply

4:13 total ; 8 stored -- 3 left at door, 1 consumed to get here, 4 cost of resupply, 5 resupply

3:20 total ; 13 stored -- 3 left at door, 1 consumed to get here, 6 cost of resupply, 10 resupply

2:33 total ; 20 stored -- 3 left at door, 1 consumed to get here, 12 cost of resupply, 17 resupply

1:33 total gold coins required in the cell

Update, further speculation is not required. The mystery has been solved! Skip to the very bottom of the post for the update. The only questions left are: First: where are the remaining coins, should they exist. We can feasibly make a claim that there may be 13 in total. Second, what is the significance of the number 26. It is speculated that is because Kingsfield, and the PS1, are now 26 years old. And finally... what happens when you keep playing and unlock the door again in the same playthrough

Preface: All uses of the word "coin" in this post are referring to Ceramic Coins, not Gold Coins which are also an item in the remake.

With the discovery of a tenth coin, the third coin in 1-1, PWWT, a lot of our original assumptions about how these coins worked have been shattered. It's about time we recompile and go over exactly what we know, and assumptions we can make on top of that.

For anyone who does not know, there is a post NewGame mode you can play on called Fractured Mode, which mirrors the entire game world, including your character and enemies. Everything. Later, it was discovered that Fractured Mode was the only way to get a unique item called Ceramic Coins, which would only appear in specific locations around the world on either Pure White or Pure Black World Tendency, on Fractured Mode. The locations and requirements are not random. A lovely commenter has posted the location spreadsheet in the comments below.

It is assumed these coins have something to do with the secret door, as what other options would we have. Properties of these coins are as follows:

Most importantly: You cannot dupe the coins. They break if placed and picked up by another player, becoming broken ceramic coins. Interestingly enough, both intact and broken ceramic coins still retain their consumable property: to regain a useless amount of health when consumed.

The coins DO carry over into New Game plus and beyond, and DO respawn. Coupled with the fact that the coins all have the exact same item value (they stack), it can be assumed that perhaps you don't even need each individual pickup location, but only need a specific number of coins.

We at first thought each individual pickup mattered because of the highly specific pattern we found the first 9 in: two per each world, one in Pure White and one in Pure Black, a highly specific pattern that would make anyone think that each and every coin was individually necessary. This was shattered when we not only found no coin in world 5 Pure White, but instead found a third coin in world 1.

Furthermore, the crows really like your coins. Give a number of coins at once to the crows, and they will react as if a successful trade had occurred, however upon leaving and reloading the world, they will still give you the coins back. Intact. This implies they want a specific number of coins. This can not be used to dupe the coins, as the crows give back the coins as a world item drop, which cannot be picked up by phantoms. They will also not accept broken coins.

Conclusions:

Coins may not be uniquely individual and might simply be a number you need to reach, perhaps through multiple NG+ runs.

They may have a consumable use aside from the pitiful health gain, much like the umbilical cords of Bloodborne, but we have absolutely zero evidence of this other than that they can be consumed.

Much more likely is that you will need a specific amount of coins to trade to the crows, and we simply need to find out what that number is. This is reinforced by the fact that the developers put an anti-dupe system into the game for the ITEM ceramic coin, which has an identical item data value to every other coin. Meaning the amount of them that you have is what's important, not what location you picked them up from.

Least likely, in my opinion, is that each individual pickup location does matter, and we will need to find each and every coin in a single world in order to get anything done, however this was a theory created when we had believed there to be a specific pattern to the coins spawning, which there is not.

I hope this covers everything that we have uncovered thus far. I really do believe the crows are the way forward. The secret door "appears locked", which implies a key. If the coins are the way forward to that key, it would logically be through the crows.

P.S. For anyone who does not know, BluePoint has precedent for using coins to unlock secrets. In their Shadow of the Colossus remake, they had a secret door which could only be opened by collecting 79 coins around the game world, behind which had a powerful weapon. It is important to note that the number 79, I believe, is significant to the game Shadow of the Colossus, and so we shouldn't immediately assume we'll need 79 Ceramic Coins for Demon's Souls.

UPDATE: This information doesn't change what our current conclusions are, but it is a nice level of coincidence to ruminate on. There are an assumed 10 coins per run, as extensive searching has not revealed an 11th coin. Each coin heals 60 HP. There are 6 difficulty cycles of new game plus's in which you can get new coins. Meaning 60 might be the number of coins we need, if these truly are hints to go by. There are also 6 knobs on the door. Also 6 streamers were tagged in that BM bluepoint tweet 2 days ago. 6 is also a colloquially demonic number, and this is demon's souls. Speaking of, originally 6 archstones.

UPDATE THIS IS NOT A DRILL DOOR HAS BEEN BREACHED. 26 COINS IS THE NUMBER. LETS GOOOOOO! If you want the pickup, it is in the comments, don't read them if you want to be surprised for yourself!!! There is the possibility of it being less than 26 coins, but the crows to not refund you for the excess, as it did not refund you 4 when 30 was spent.

Important to note that the reward cannot be dropped and is locked to the save in which you acquire it, so every build that uses the armor will need to currently go to NG+2. Hopefully we'll have found 13 coins soon enough.

Also there is apparently an 11th coin found, check the list for updates.

https://www.reddit.com/r/demonssouls/comments/juvn5l/spoilers_remake_discovery_megathread_13_mystery/gct18bo?utm_source=share&utm_medium=web2x&context=3