Solving this puzzle requires three steps:

1. Figuring out which vote-counting method is being used to determine each of the three winners
2. Using that knowledge, fill in the empty spaces on the ballots
3. Extract the answer

1: Figuring out vote-counting methods

There are many ways to complete this part. By constructing ballots and scoring these mock elections, you can determine that the three winners are determined using three classic counting methods:

• Winner #1 is using a Borda Count. This method scores each ballot, giving 5 points to the first candidate, 4 points to the second, 3 to the third, 2 to the fourth, and 1 to the fifth. The candidate with the most total points wins.
• Winner #2 is using Instant Runoff Voting. This is an iterated method where in each iteration, the candidate(s) with the fewest first-place votes are eliminated, and all ballots are adjusted to remove those candidates. The process repeats until only one candidate remains.
• Winner #3 is using First-Past-The-Post. In this method, the winner is the candidate with the most first-place votes.

2: Filling in the ballots

This stage is a straightforward logic puzzle. There's only one way to fill in the ballots so that the winners match the given winners. Here's the solutions for the blank spaces:

George (6)
Donald (5)

Donald (3)
George (2)

George (3)
Rutherford (4)

Donald (2)
Rutherford (6)
John (3)

Donald (2)
John (2)

Benjamin (1)
George (6)

Rutherford (5)
John (2)

Donald (4)
Rutherford (5)
George (6)

George (6)
John (5)

George (5)
Rutherford (1)

You can check this solution by scoring the filled-in ballots as a mock election, and verifying that the results of the mock election match the actual result.

3: Extraction

At this point, you may notice that the given indices cannot be used to index into the names (because index 5 is given for John). Instead, notice that the five names (George, Rutherford, Donald, John, and Benjamin) are the names of the five presidential candidates who won the election, but did not win the popular vote: George W. Bush, Rutherford B. Hayes, Donald Trump, John Quincy Adams, and Benjamin Harrison. Substituting their names for the names of the candidates who won the popular vote in that election (respectively: Albert Gore, Samuel Tilden, Hillary Clinton, Andrew Jackson, and Grover Cleveland), we get the following names and indices:

• Albert (6)
• Hillary (5)
• Hillary (3)
• Albert (2)
• Albert (3)
• Samuel (4)
• Hillary (2)
• Samuel (6)
• Andrew (3)
• Hillary (2)
• Andrew (2)
• Grover (1)
• Albert (6)
• Samuel (5)
• Andrew (2)
• Hillary (4)
• Samuel (5)
• Albert (6)
• Albert (6)
• Andrew (5)
• Albert (5)
• Samuel (1)

Indexing into these substituted names gives the clue phrase TALL BUILDING TEN LETTERS, which clues the answer, SKYSCRAPER.