First, you solve the puzzles, getting 15 answers...
(ARAUCARIA, BITMAP, CUFFLINK, GLYPHS, KEEP TIME, LIBRARY, SHRIMP, SILLY BLURB, SYLLOGISM, SYMBIOSIS, TUXEDO, WALKY TALKY, WEEKEND, ZUCCHINI, ZURICH)
...in addition to 5 answers that are already filled in on the mysterious icosahedron.
(DINGHY, EGO, FLAPJACK, JUNGLE, TOOLBOX)
The 15 puzzle answers go in the blanks on the faces of the icosahedron, constrained by their lengths. As you do so, you spot another constraint: for each vertex of the icosahedron, the five answers adjacent to that vertex share a letter (which is not contained in any other puzzle answers).
This puts a letter at each of the vertices, and we finally get to use the last feature of the icosahedron: the black lines along some of the edges, which connect the vertices into one path. Reading the letters along the path spells out GOBACKTENHRS: instructions to use the icosahedron as a time machine that will take campers back to before sign-in ends.