Goomba structure
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A Goomba Structure is created by mass cloning a large number of Goombas, placing them in desirable positions for Mario to utilize them. This can only be done in stages where cloning is possible and Goomba spawners are present.
As Mario is unable to release the clone normally without interacting with the Goomba's hitbox, numerous glitches that make Mario release the clone remotely is required. Hat-in-hand glitch, for instance, allows Mario to release Goombas at the HOLP since the object in front of Mario does not render, leaving the HOLP unaltered. Pause Buffered Hitstun allows Mario to release a Goomba clone since Mario is invisible in every other frame.
Goomba Cluster
A Goomba Cluster is created by releasing cloned Goombas in different angles at the same HOLP. Because the Goombas are facing different directions, they have slightly different hitboxes, allowing them to be individually stomped.
Goomba Clusters are typically used to build Goomba Bridges and Goomba Staircases, both of which require a staircase of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} to set the HOLP to where Goomba Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n+1} must be placed. Without a Goomba Cluster, an entire bridge of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} must be built twice to build a bridge of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n+1} , causing both time and the number of slots to grow with order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2^N} .
However, with a Goomba Cluster, all of the Goombas at position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} can be placed once, removing the need to keep setting the HOLP. Each Goomba position on the bridge still requires a HOLP update, however, the HOLP must only be set there once. Therefore, at position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} , the number of Goombas required is the number of Goomba positions after that position, which is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N+1-n} .
Since there is a position containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n} Goombas for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\in [1,N]} , the total number of Goombas required is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {N(N+1)}2=\frac {N^2}2+\frac N2} , something that grows much more efficiently on the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N^2} .
Goomba Bridge
A Goomba Bridge is a type of structure created by cloning Goombas, allowing Mario to navigate in the air to certain locations that are hard to reach. The bridge can be as small as just one Goomba, to a massive bridge involving Goombas of two digits. Creating a bridge allows Mario to move across certain gaps that would otherwise require long jumping or triple jump dives.
The most efficient way to create Goomba Bridges is setting up the HOLP first and then release Goomba clones remotely at different angles to create a Goomba Cluster, and then step on that cluster to set a different HOLP further away, and then repeat. Before the discovery of Goomba Clusters, the old method involves cloning the Goombas one by one, which requires many trips back and forth and takes an exponential amount of Goombas to complete.
Goomba Staircase
A Goomba Staircase is similar to a Goomba Bridge, except that the main purpose is for Mario to gain height instead of crossing a gap. Massive staircases can be constructed for the purpose of collecting the items to finish the objective high in the air, such as the secrets in Mario Wings to the Sky, or to reach the higher platforms in Bowser in the Sky.
Creating Goomba Staircases is most efficiently done by setting up the HOLP and create a Goomba Cluster for each step.
Goomba Oasis
A Goomba Oasis is a type of flat structure created by cloned Goombas via transport cloning. The clones are located midway between Mario and the HOLP laterally and at the HOLP's height. This allows many Goombas clones to stay in midair without having to construct a massive Goomba Bridge, as the game has limited memory for the object slots.
Goomba Ladder
A Goomba Ladder is a structure comprising of cloned Goombas going straight up, forming a ladder. The name is often used for any Goomba structure, but this use is a misnomer.
It can be created by releasing the cloned Goomba by dropping instead of throwing, making them appear at Mario's height but at HOLP laterally. This allows Mario to bounce on each Goomba to gain height. The disadvantage is that the ladder can only be as high as the course allows it, since the Goombas cloned with this method cannot go above the height that Mario can stand on at any point. Currently, there is no use of this structure in challenges.