Finishing each game in the Ninteamdo Playmate adds three badges to the badge collection. After finishing the three world metas, numbers appear on the badges when they are inspected.
This puzzle is a metameta, meaning it uses the answers from the three rounds on the world map. In total, there are 24 puzzles in these rounds, and each one corresponds to one of 24 badges. Each badge has 5 numbers — one in the center and one on each of the sides.
The numbers on the sides are perfect fourth powers, hinting that this puzzle relates to something four dimensional. By using the fourth root, we get numbers that can be indexed into the answer corresponding to each badge. (We are uncertain at this point about the purpose of the numbers in the center, which are neither fourth powers nor fit as indices into the corresponding answers.)
We have 24 square badges, and with a bit of searching, we find that the tesseract (4-dimentional hypercube) has 24 faces. Another clue that the tesseract is the object of interest here is that the letters for the badge sides come in multiples of 3. (This should be apparent even when teams are missing a few answers / badges.) In the tesseract, every edge is shared by 3 different faces. At this point, we realize that we need to build a tesseract so that the edges match.
A lot of edges are unique -- every letter that only appears 3 times must be put on the same edge. Some teams used the fact that the sequence of badges form a path on the tesseract to help place the faces, an assumption that is correct but not necessary to form the tesseract. By building it out, we end up with a tesseract like this.
The badges are numbered, indicating that they form a sequence. On the tesseract, this forms a traversal along adjacent faces (aka a Hamiltonian Path over the faces). If we trace out this path, along the edges, we get READ OUT THE OPPOSING FACES. (This message can also be extracted without building the tesseract and just noticing that consecutive badges must share a letter, but teams can't do the next step without actually building the tesseract.)
This clues us to look at the opposing or antipodal face when tracing out the path. (A face's opposing face is parallel to it but differs from it in both orthogonal axes. For example, if modelling a tesseract as an inner and outer cube, the opposing face of the front face of the outer cube is the back face of the inner cube.)
We now notice that the numbers in the center of each badge can index into the answer for the opposing face. If we traverse the same path and index the center number into the opposing answer (or alternatively traverse the opposing path and index the center numbers of the original path into the answer), we get the final answer TEAMMATES IN THE GAME OF LIFE.
The number 24 holds a special meaning to teammate, being the number of permutations of TEAM. When we started, we thought it would be neat if our hunt structure made use of this. Though we rejected stronger integrations (I didn't want to have to come up with EMAT, AMTE, or EAMT puzzles), in the end we had this 4D construction puzzle combining 24 faces.
I realized after coming up with the idea that the last step was similar to the 2018 Mystery Hunt Catan Meta, but I think on the whole, Badge Collection feels quite different. Besides, we have another whole dimension 😛. Throughout the testsolves and during the hunt, it has been incredibly exciting to see all the different ways solvers have come up with to represent a 4D object on a 2D screen, or in some cases, even 2D paper.
A week before our hunt, we almost pivoted and made the final answer about a COMPANION TESSERACT to reduce anagramming potential (indices used to work slightly differently), but we were all divided on this and Bryan came in with a clutch save that worked with the original answer. Sorry if anyone saw the initial cake picture and was expecting the entire hunt to be a Portal reference.