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The Blue Carr Bungle

Kevin Wald

Each of the "Locked Rooms" corresponds to the unit cell of one of the minerals the Mohs scale is defined in terms of. (Various hints in this direction:

In particular, each Locked Room has the shape of the corresponding unit cell (using a hexagonal, rather than rhombohedral, unit cell for the trigonal system, as the bottom parenthetical remark indicates), and the edge lengths shown are the same as the lengths of the cell edges in angstroms, except that one indicated length (shown with its integer part in orange) has been decreased by a whole number. (This is hinted at by the "falls this far short with regard to math, right at the start" part of the flavortext.) From the outside in, we have:

Fluorite: Isometric; a = 5.463, shown as 1.463 (4 less).
Corundum: Trigonal; a = 4.751; c = 12.97, shown as 7.97 (5 less).
Apatite: Hexagonal; a = 9.38, shown as 2.38 (7 less); c = 6.89.
Topaz: Orthorhombic; a = 4.35; b = 8.8; c = 8.4, shown as 5.4 (3 less).
Calcite: Trigonal; a = 4.989; c = 17.062, shown as 14.062 (3 less).
Gypsum: Monoclinic; a = 5.68; b = 15.18, shown as 10.18 (5 less); c = 6.29.
Quartz: Trigonal; a = 4.9133, shown as 0.9133 (4 less); c = 5.4053.
Diamond: Isometric; a = 3.5668, shown as 0.5668 (3 less).
Orthoclase: Monoclinic; a = 8.625, b = 12.996, shown as 5.996 (7 less); c = 7.193.

Using the differences as indices into the mineral names gives ONE PLURAL. The mineral that defines 1 on the Mohs scale (and the only one of the 10 minerals defining the Mohs scale that is not represented in the diagram) is talc; the answer is the plural, TALCS.