### 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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## Comments

Scary math. Lost me after the title, but then I am not the sharpest of knives in the rack!

Why is voting easy negative?

I can only leave this comment to this teaser &#@!^*)(&*%@#$%^*? I'm sure it was a good teaser for those who undestood it

I get how it works but very hard to analyse a regression when the results have been rounded off. Probably needs more time than I'm prepared to give it.

jimbo - One way to try and figure out fractions when the result has been rounded is to look at the convergents to a continued fraction with the rounded value. This web page: http://www.mcs.surrey.ac.uk /Personal/R.Knott/Fibonacci/cfCALC.html (remove spaces from url before using) has a continued fraction calculator that can be helpful. For example if you give it 2.29 on the left, you get convergents: 2, 7/3, 16/7, 71/31, and 229/100. The first of these that rounds to 2.29 is 16/7, and that also fits the context of the problem best. Try it with some of the other numbers in the problem!

This was not all that difficult. Even rounded off to 2 fractional digits, it's easy to recognize thirds and sevenths, etc. But I am a little confused, I think maybe the rating system has been altered since this teaser was written..? Because, it seems to imply that there are only two options for voting, positive and negative. But now it's 0 (negative), 1/2 (neutral), and 1 (positive). Probably the same formula, I'm sure, just with 1/2's counting as 0.5 positive votes. Just like draws in a winning percentage of a team, of course. Good teaser! Don't worry about the ones saying it's too hard.

Thanks, eighsse. Yes, as you can tell from the dates of the first comments, this teaser goes back several years to when there was a simpler rating system in place. Thanks for your interest.

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