AnNi2020 - The hunt for Darvin

[Demonic God]

On a beautiful day, Darvin had surfed into a brand new region of the realm, one that the realmshapers were working on. It is detrimental however, due to the cryptic nature of a place with such intense creation magic, that it served as a prison to poor ol Darvin, in such that he can't escape on his own

Mur then tasked the realmshapers to go after Darvin, and bring him back to safety

The new region is under construction, and take the form of a large tower consisting of 3 floors. Each floor is essentially a square of 3x3, 5x5 and 9x9 rooms respectively. Darwin starts on the top floor, and needs to be banished one by one, eventually out of the tower and back to the realm.

Each turn, a realmshaper can teleport to a single room, and should they find Darvin there, banish him to a random room on the floor below, or out of the tower if he is on the lowest floor.

Each turn, after the realmshapers had teleported, Darvin will surf the heat vein and move to a random adjacent (sharing a wall - NSEW) room.

Do note that even if Darvin chance upon a realmshaper, due to the unfinished nature of the realm, the shall not be able to see him, unless they refresh their vision, which alas, takes the same amount of time as it would for them to move to another location. Thus, put simply, after Darvin's move, he could only be seen or captured should a shaper decided to stay within the room, or came into the room from elsewhere.

What is the minimum amount of realmshaper needed on each floor, to guarantee Darvin's capture?

Details:

• There exist a NxN grid, in which Darvin may be in any square
• Each turn, you can select some squares. If you selected the square Darvin is in, you "win"
• If you didn't find him, Darvin will move 1 square NSEW (cannot go over walls for obvious reasons)
• What's the lowest number of squares you need to select per turn to guarantee that you will catch Darvin eventually?

Rewards:

• Tier 1: solving the 3x3 floor - 5 plushies
• Tier 2: solving the 5x5 floor - 5 plushies and a gold coin
• Tier 3: solving the 9x9 floor (or general case) - 5 plushies, a gold coin and anni crit (first 3 solvers - 2 gold coins afterwards)

Note: Rewards are not final, and may be bumped once I can establish how difficult others find this challenge. A correct proof is required. Anni crit can be given out as tier 2 if too few were able to solve tier 3. Tier 3 may hopefully be bumped to reward a WP if deemed difficult enough.

Edited April 14, 2020 by Demonic God
clarification

[Demonic God]

Here shall be the list of all participants that had submitted a solution and their current progress:

Aia: Solved the top (3x3) floor. It appeared that I had rushed the submitted solution and was wrong. Partly due to my own phrasing, which I had fixed, and a detailed non-MD description shall be added shortly 
Chewett: A simple general solution (not optimal), and a decent 3x3 solution (not optimal)
Steno: A decent general solution
Aelis: A simple general solution, and the optimal solution for 3x3. Congratulations on being the first

Edited April 21, 2020 by Demonic God

[Aia del Mana]

This is an amazing quest to ponder upon. I have not yet completed it (and fear that I may not be able to this anniversary owing to time constraints; still, I will continue to ponder on it following.)

If naught do sponsor a wish-point for a general solution that satisfies all sizes of grid 3x3 or greater (the "general case"), then I shall do so myself; this quest, I believe, should become a permanent one.

[Tissy]

Could the realmshaper teleport to random room within one floor or cross floors?

Basically the realmshaper cannot see Darvin unless he teleport to the room where Darvin is right?

[Demonic God]

1 hour ago, Tissy said:

Could the realmshaper teleport to random room within one floor or cross floors?

Basically the realmshaper cannot see Darvin unless he teleport to the room where Darvin is right?

Each floor is a separate problem with a separate solution. 

[Demonic God]

Given that I recently raked in some silvers from @Ungod (bless him for that xD), I'll sponsor extra rewards for all who submitted a good attempt at solving the quest, but did not get a correct answer!

The higher tier you go, the more reward you get! The only stipulation is that you need a strategy that works, and describe in detail of how it would guarantee Darvin's capture! Rewards are based greatly on the beauty and elegance of your solution, and you need not prove if it is optimal. Multiple submission is allowed, but you are only rewarded for the best solution (and you will be paid the difference from your previously claimed reward)

Rewards will range from 1-6 SC, based on the tier you solved and the elegance of your solution!

Based on this new criteria, Steno had won 3 SC for his attempt, and Chewett 1SC. Please contact me in-game

With this new reward implemented, I would also like to state again, that a rigorous proof would be required to win the official prize. It is rewarded for finding and proving the optimal strategy, not simply finding it. I will sponsor additional reward, should you stumble upon this strategy, but couldn't prove it optimal.

Edited April 19, 2020 by Demonic God

[Tissy]

It deserves at least one wish point, no, two!

[Demonic God]

Aelis was the first to find the optimal solution for 3x3! Congratulation!
 

[Steno]

Only downvoting because I didn't win. And I want to KNOW the answer!

-not actually mad

 

Got rid of downvoting, I don't want to give you(DG) any bad rep because of me being funny

Edited April 21, 2020 by Steno

[Demonic God]

I guess I need to summarize this too.

Aelis is the only successful solver of the 3x3 during the anni. Congratz

I'll be trying to make this quest official, and earn some better rewards for it permanently. For now, I will try to reward all solver of tier 1, 2 and 3 with 8 SC, 1 GC and 2 GC respectively.