Triangles

The 20 triangles can each be solved with answers corresponding to the 60 given lengths. (See the PDF in the appendix.)

These answers fall into 12 groups of 5 answers each:

• Keys: ALT, CAPS LOCK, RETURN, SHIFT, TAB
• Letters: V, W, X, Y, Z
• Shakespeare's Julius Caesar: DOGS OF WAR, ET TU, GREEK TO ME, IDES OF MARCH, UNKINDEST CUT
• Majors: ANTHROPOLOGY, ARCHITECTURE, HISTORY, LITERATURE, PHYSICS
• Birds: EMUS, HENS, HERONS, PELICANS, SWANS
• Monopoly: ELECTRIC COMPANY, FREE PARKING, GO, INCOME TAX, WATER WORKS
• NATO alphabet: BRAVO, ECHO, INDIA, MIKE, NOVEMBER (phonetic alphabet)
• First names of '90s comedic movie title characters: ACE, AUSTIN, BILLY, DEUCE, HAPPY
• Nutrients: FOLATE, MAGNESIUM, POTASSIUM, THIAMIN, VITAMIN C
• Dutch Eredivisie: AJAX, ENSCHEDE, GO AHEAD, NEC, ROTTERDAM
• Red Sox players: BABE, PESKY, RICE, SPEAKER, YOUNG
• Celestial objectsTHE GALAXY, THE MOON, THE PLANETS, THE STAR, THE SOLAR SYSTEM

This allows the 20 triangles to be cut and assembled into an icosahedron, with each group of 5 touching at a common vertex. The appendix PDF has the triangles assembled in a plane such that they could be folded.

Once we have done so, we notice that one answer on each face of the icosahedron associates with a unique number from 1 to 20:

1. ACE (playing card)
2. DEUCE (playing card)
3. HENS (12 days of Christmas)
4. ARCHITECTURE (MIT course number)
5. V (Roman numeral)
6. PESKY (jersey number)
7. SWANS (12 days of Christmas)
8. PHYSICS (MIT course number)
9. TAB (ASCII code)
10. X (Roman numeral)
11. NOVEMBER (month)
12. MAGNESIUM (atomic number)
13. RETURN (ASCII code)
14. RICE (jersey number)
15. IDES OF MARCH (actual day)
16. INCOME TAX (16th amendment)
17. THE STAR (major arcana)
18. THE MOON (major arcana)
19. POTASSIUM (atomic number)
20. ENSCHEDE (in the Twente region ~ 20)

These allow us to label the icosahedron as a d20. As confirmation, opposite sides of the die add up to 21, as is standard.

Finally, we can use the D&D campaign rules. Generating a random rule produces one of 20 rules, with unequal frequencies. Additional inspection will find that they appear with a frequency ratio of 1:2:3:...:19:20 (exact for the first 210 rules generated, with those odds afterwards), providing a mapping. This allows us to map each rule onto a specific face of the d20.

These rules correspond to five major editions of the Dungeons and Dragons rulebooks, with the relevant edition names provided (see Editions of Dungeons & Dragons - Wikipedia): AD&D 1, D&D Basic BECMI, AD&D 2, D&D 3, D&D 5e. Each rule is only valid in certain systems.

Treating each rulebook separately, painting the indicated faces produces a letter on the constructed icosahedron, as above. (The appendix PDF has the planar versions of the icosahedron, with the triangles arranged and shaded so that it is possible to see what letter is intended.) In order by rulebook, these spell out the answer, DISCS.

Appendix