FizzBuzz Ruby one-liner

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Rosettacode.org has this excellent one-line FizzBuzz solution in Ruby.

1.upto(100){|n|puts'FizzBuzz '[i=n**4%-15,i+13]||n}

The trouble is, I don’t understand it. The part that puzzles me is the ”n to the power of 4 modulo -15”. Does anyone have an explanation or a reference to an explanation? I want to use this way of selecting substrings in other problems. For more information on FizzBuzz, see [https://rosettacode.org/wiki/FizzBuzz]

I don't know how they discovered to raise to the fourth power, but the -15 is because FizzBuzz deals with multiples of 3 or multiples of 5 or multiples of both 3 and 5 (ie, multiples of 15)...then negating it ends up working with negative indices quite well. We can see that it works with Modular Exponentiation. The Memory-efficient method section there says:

c mod m = (a ⋅ b) mod m
c mod m = [(a mod m) ⋅ (b mod m)] mod m

In our case, the c is our n, so we have

c ** 4 % m

using the law of exponents, we know that (c ** e1) * (c ** e2) = c ** (e1 + e2), so c ** 4 = (c ** 2) * (c ** 2), so we now have an a and a b, which are both c ** 2. Thus:

(c ** 4) % m = ((c ** 2) * (c ** 2)) % m              = (((c ** 2) % m) * ((c ** 2) % m)) % m              = (((c ** 2) % m) ** 2) % m

and following the same steps, again:

(c ** 2) % m = (c * c) % m              = ((c % m) * (c % m)) % m              = ((c % m) ** 2) % m

and finally:

(c ** 4) % m = ((((c % m) ** 2) % m) ** 2) % m

When m = -15, the only values for c % m are (-14..0) and we can build a simple table to look at. Since we only ever operate on the result of a modulo, we only need to be able to prove these 15 numbers work:

c%m    **2     %m    **2     %m -14 => 196 => -14 => 196 => -14 -13 => 169 => -11 => 121 => -14 -12 => 144 => -06 =>  36 => -09 -11 => 121 => -14 => 196 => -14 -10 => 100 => -05 =>  25 => -05 -09 =>  81 => -09 =>  81 => -09 -08 =>  64 => -11 => 121 => -14 -07 =>  49 => -11 => 121 => -14 -06 =>  36 => -09 =>  81 => -09 -05 =>  25 => -05 =>  25 => -05 -04 =>  16 => -14 => 196 => -14 -03 =>   9 => -06 =>  36 => -09 -02 =>   4 => -11 => 121 => -14 -01 =>   1 => -14 => 196 => -14  00 =>   0 =>  00 =>   0 =>  00

Now, looking at our table, the values for all multiples of 3 are -09, the values for all multiples of 5 are -05, and things that are multiples of 3 and 5 are set to 00; everything else is -14 (If we had used 15 instead of -15, we'd have 6, 10, 0, and 1, respectively, and would need a look up to turn that into string indices). Plugging those in for the start parameter of String#[] with the string 'FizzBuzz ' gives us:

'FizzBuzz '[-9] # => 'F' 'FizzBuzz '[-5] # => 'B' 'FizzBuzz '  # => 'F' 'FizzBuzz '[-14]# => nil

and adding 13 to those numbers to get the length:

'FizzBuzz '[-9, 4]   # => "Fizz" 'FizzBuzz '[-5, 8]   # => "Buzz " 'FizzBuzz '[0, 13]   # => "FizzBuzz " 'FizzBuzz '[-14, -1] # => nil