PLEASE NOTE! A different problem that uses this same ruleset that can likely only be solved with brute force and a computer program has been posted to math.se, viewable HERE. I was unaware of the crossposting etiquette on stackexchange, my bad comrades!
THE GAME DEFINED
- The game takes place on a board with a square grid. For the purposes of this puzzle, you can assume the board size is infinite.
- The board's grid squares are all in a "blank state" at the start of the game.
- Clicking (or tapping, etc.) a blank grid square places a tier1 tile.
- When three tier1 tiles are found to be adjacent in any of the 4 cardinal directions, a tier2 tile is placed at the site of the tier1 tile that was most recently placed. The other two tier1 tiles that formed the chain of three are reverted to blank grid squares.
- An extension of the previous rule: three adjacent tier2 tiles create a tier3 tile, etc. Tier5 tiles are the greatest tile possible (see the postulates section for the reason why).
- The edges of the game board (not relevant in this problem) and all non-blank grid squares are considered "walls". No tile can exist on top of or inside a wall; this game exists on only one "gameObject layer" (shout-out to people whom've used Unity for game making!).
- Tier2 tiles cannot be created adjacent to 4 walls. Only tier1 tiles have the property of being able to be placed on any blank grid square.
- Tier3 tiles cannot be created adjacent to 3 walls.
- Tier4 tiles cannot be created adjacent to 2 walls.
- Tier5 tiles cannot be created adjacent to any walls. Tier6 tiles are impossible to create because of this: due to the rule that all non-blank grid squares are considered walls, a Tier5 tile cannot be created adjacent to any other tile, and therefore cannot be arranged into a chain of three adjacent tiles to reach the next tier.
Important to note!
- In this puzzle, what we care about are your unique grid square clicks. Because the least recent two tiles that formed a higher tier tile are reverted back to blank grid tiles, it's possible to click a specific grid square multiple times in the creation process of any tile tier3 and above. Unique grid square clicks are defined as the complete set of board squares that you clicked at least once, even if they end up as blank tiles at the end of your solution.
THE PROBLEM EXPLAINED
Earlier today I posted this question on the mathematics stack exchange. While it hasn't received a proper answer, StackExchange user Servaes offered this formula for calculating the minimum number of unique grid square clicks needed to create a tile of a given tier:
"The minimum amount of grid squares required to reach a $T_n$ tile is . . . $T_n=2n-1$"
According to this proposed formula, a tier4 tile can be made with 7 unique grid square clicks (27 clicks total), and a tier5 tile can be made with 9 unique grid square clicks (81 clicks total). While I agree it's possible to create a tier4 tile in 7 unique grid square clicks, I disagree that it's possible to make a tier5 in 9.
And so I pose this conundrum to you:
What is the fewest amount of unique grid square clicks needed to create a tier5 tile? Please show your work: I want to understand the shape formed by your unique grid square clicks, and I want to know the chronological order and coordinates of your clicks (81 clicks is a lot to give coords for, so feel free to only give the order and coords of your t4 tiles (3 total ordered coords), t3 tiles (9 total ordered coords), or t2 tiles if you're feeling spicy (27 total ordered coords)).
While it's explaining the separate problem I linked earlier in this post, and while it also contains potentially confusing terms that are not relevant to this specific puzzle, I have created a YouTube video that may help you grok the game I've described here. Click this link to check it out!