### Brain Teasers

# Reverse Engineering

While rating some Braingle teasers recently, I started taking notes on how my

vote affected the ratings. Some of my data is presented below. Can you figure out what is the likely formula that computes the ratings from the votes? Can you determine for each teaser how many votes were cast for each attribute prior to mine, and how many of those were positive (fun/hard)? Remember, not every voter necessarily votes for both attributes.

Notation: The notation for Teaser A below shows that it started with a fun rating of 2 and difficulty 1.33, based on 6 voters, that I voted positively on the first attribute (fun) and negatively on the second (easy), and that it wound up with ratings of 2.29 and 1.14, respectively, after my vote.

Teaser A: (2.00, 1.33)[6] (+, -) -> (2.29, 1.14)

Teaser B: (1.14, 0.67)[7] (-, -) -> (1.00, 0.57)

Teaser C: (3.00, 0.67)[8] (+, +) -> (3.11, 1.14)

Teaser D: (3.11, 0.89)[9] (+, -) -> (3.20, 0.80)

Teaser E: (2.33, 2.46)[13] (+, -) -> (2.46, 2.29)

vote affected the ratings. Some of my data is presented below. Can you figure out what is the likely formula that computes the ratings from the votes? Can you determine for each teaser how many votes were cast for each attribute prior to mine, and how many of those were positive (fun/hard)? Remember, not every voter necessarily votes for both attributes.

Notation: The notation for Teaser A below shows that it started with a fun rating of 2 and difficulty 1.33, based on 6 voters, that I voted positively on the first attribute (fun) and negatively on the second (easy), and that it wound up with ratings of 2.29 and 1.14, respectively, after my vote.

Teaser A: (2.00, 1.33)[6] (+, -) -> (2.29, 1.14)

Teaser B: (1.14, 0.67)[7] (-, -) -> (1.00, 0.57)

Teaser C: (3.00, 0.67)[8] (+, +) -> (3.11, 1.14)

Teaser D: (3.11, 0.89)[9] (+, -) -> (3.20, 0.80)

Teaser E: (2.33, 2.46)[13] (+, -) -> (2.46, 2.29)

### Hint

Try to recognize common fractions represented by the decimalpart of each rating. Also: remember that the ratings range from 0 to 4.

### Answer

The apparent formula, for p positive votes out of v cast, is 4*p/v. The following lists the fractions that gave rise to the data above. To get the number of positive votes, divide the numerators by 4.Teaser A: (12/6, 8/6)[6] (+, -) -> (16/7, 8/7)

Teaser B: (8/7, 4/6)[7] (-, -) -> (8/8, 4/7)

Teaser C: (24/8, 4/6)[8] (+, +) -> (28/9, 8/7)

Teaser D: (28/9, 8/9)[9] (+, -) -> (32/10, 8/10)

Teaser E: (28/12, 32/13)[13] (+, -) -> (32/13, 32/14)

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