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What is the two word answer to the puzzle below?

16777215+262709978263278+3435973836 = (4)

281474976710655+1240612796065921481376886589594988270+967140655691703339764940 = (12)

1208925819614629174706175+1240612796065921481376886589594988270+225179981368524 = (11)

65535+15658734+52428 = (11)

4294967295+262709978263278+13421772 = (10)

(4)+(12)+(11)+(11)+(10)= (8)(6)

Hint 1

One of the three is a synonym, but overall this is just a regular puzzle.

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    $\begingroup$ Damn, got the first bit fairly quickly and been stumped ever since, very cool idea first idea though! $\endgroup$
    – Beastly Gerbil
    Commented Oct 14, 2023 at 13:47
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    $\begingroup$ @BeastlyGerbil Keep going :) Try to include the rebus and knowledge tags somehow $\endgroup$
    – Prim3numbah
    Commented Oct 14, 2023 at 13:55
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    $\begingroup$ I just noticed 65535 and now I'm suspicious… $\endgroup$
    – Someone
    Commented Oct 14, 2023 at 18:04

1 Answer 1

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This puzzle is about ...

... the five Platonic solids, which is also the overall two-word answer.

The first step ...

... is to write the long numbers in hexadecimal:

FFFFFF₁₆ + EEEEEEEEEEEE₁₆ + CCCCCCCC₁₆ = (4)
FFFFFFFFFFFF₁₆ + EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE₁₆ + CCCCCCCCCCCCCCCCCCCC₁₆ = (12)
FFFFFFFFFFFFFFFFFFFF₁₆ + EEEEEEEEEEEEEEEEEEEEEEEEEEEEEE₁₆ + CCCCCCCCCCCC₁₆ = (11)
FFFF₁₆ + EEEEEE₁₆ + CCCC₁₆ = (11)
FFFFFFFF₁₆ + EEEEEEEEEEEE₁₆ + CCCCCC₁₆ = (10)

That's a bit repetitive, so let's clean that up:

 6·F + 12·E +  8·C = (4)
12·F + 30·E + 20·C = (12)
20·F + 30·E + 12·C = (11)
 4·F +  6·E +  4·C = (11)
 8·F + 12·E +  6·C = (10)

(That's nonsense from a mathematical point of view, of course, but I'm applying the magic of the rebus tag here.)

What does that mean?

Each line has a number of Fs, Es and Cs. Also, only certain numbers appear: 4, 6, 8, 12, 20 and 30.

It took me a while to see this, but the lines refer to the Faces, Edges and Coordinates (vertices) of the five Platonic solids:

cube: 6 faces, 12 edges, 8 vertices
dodecahedron: 12 faces, 30 edges, 20 vertices
icosahedron: 20 faces, 30 edges, 12 vertices
tetrahedron: 4 faces, 6 edges, 4 vertices
octahedron: 8 faces, 12 edges, 6 vertices

The numbers in parentheses are the letter enumerations for these words. The overall two-word answer has (8) and (6) letters: Platonic solids.

(The Platonic solids are the only five convex regular polyhedra, so that's what the "regular" in the hint refers to. "Vertex" requires a synonym that starts with one of the letters used for hexadecimal notation, but not E or F.)

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    $\begingroup$ Ahhh very nice, interesting bit of trivia too - the numbers in the puzzle are known as 'Brazilian Numbers' - numbers that are repdigits in another base $\endgroup$
    – Beastly Gerbil
    Commented Oct 16, 2023 at 12:03
  • $\begingroup$ Well done! You got the correct answer. Just check the terms in row 2 and row 3 under "the first step". Everything else looks perfect! $\endgroup$
    – Prim3numbah
    Commented Oct 16, 2023 at 12:20
  • $\begingroup$ Oh, sorry, that was an old draft, where I (well, the program I used) stopped at 64 bits ... :( (And I'm still missing a link to the title.) $\endgroup$
    – M Oehm
    Commented Oct 16, 2023 at 12:30
  • $\begingroup$ No prob, looks good now. $\endgroup$
    – Prim3numbah
    Commented Oct 16, 2023 at 12:34
  • $\begingroup$ Might be worth noting the pun in the hint... $\endgroup$
    – Gareth McCaughan ♦
    Commented Oct 21, 2023 at 21:57

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