#How are you supposed to solve a pattern of numbers?

10 messages · Page 1 of 1 (latest)

chilly thorn
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An example could be ' 2 9 28 65 ? 217 ', which in this case the answer should be 126.
The rule in this case would be 1^3+1, 2^3+1, 3^3+1, 4^3+1 etc. These rules seem hard to find and I was wondering if there was a way to solve these questions without blindly testing rules.

queen pawnBOT
hot badge
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First, there is no solution!!! The are an infinite number of sequences that contain these data points.

We're really looking for the simplest to describe sequence that explains the data.

n^3+1 is the ONLY order 4 or less polynomial that fits all the given terms so it's a pretty simple description and obvious first guess (at least to a computer)

https://en.wikipedia.org/wiki/Lagrange_polynomial

📎 polyFit2.py
Lagrange polynomial

In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data.
Given a data set of coordinate pairs

    (
    
      x
      
        j
      
    
    ,
    
      y
      
        j

...

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If you don't have a computer,

  1. you can look at d(n) = a(n)-a(n-1) and see that the sequence is super linear.
  2. you can look at d(n)-d(n-1) and see that the sequence is super-quadratic
chilly thorn
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Alr damn

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lofty sapphire
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@chilly thornit's like this

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you continue until it's the same number, then just continue the patterns going back up

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it would work even for something like 6x³ - x + 12