In Facebook, Xing, Google+, Vk.com, Linkedin and other of these social media networks we are often encountered with a trivial question like this:
1->2
2->8
3->18
4->32
5->50
6->72
7->?
There are some easy patterns. Either it is some polynomial formula or some trick with the digits.
But the point is, that any such sequence can easily be fullfilled by a polynomial formula. That means we can put any value for 7 and make it work. Or any answer is correct. So what would probably be the real question is the most simple function to full-fill the given constraints. Simplicity can be measured in some way… If the solution is unique is unclear, but let us just look at the polynomial solution.
A function is needed that takes as parameter a list of key-value-pairs (or a hash map) and that yields a function such that the function of any of the key is the associated value.
Assuming a polynomial function in one variable we can make use of the chinese remainder theorem, which can be applied to univariate polynomials over a field 
as well as to integral numbers. For a polynomial p(X) we have
![\[p(x) \equiv p(X) \mod X-x\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-11f05dcab8fa434834fad1be86aca103_l3.png "Rendered by QuickLaTeX.com")
where 
is the polynomial variable and 
is a concrete value.
We are looking for a polynomial 
such that for given values 
we have
![\[\bigwedge_{i=0}^{n-1} p(x_i) = y_i\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-2e65963ed4a7ddd649bc5e56f341b57e_l3.png "Rendered by QuickLaTeX.com")
or in another way
![\[\bigwedge_{i=0}^{n-1} p(X) \equiv y_i \mod X-x_i\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-d511ee0bac00174e402f48f61b0226b3_l3.png "Rendered by QuickLaTeX.com")
which is exactly the Chinese remainder theorem.
Let
![\[I=\{0,\ldots,n-1\}\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-51196df526f571ed87ffc567700f3a5f_l3.png "Rendered by QuickLaTeX.com")
and
![\[\bigwedge_{j=0}^{n-1} I_j = I \setminus \{j\}\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-abece6459cd49492382497098b829e71_l3.png "Rendered by QuickLaTeX.com")
We can see that for all 
the polynomials
![\[e_i = \prod_{j \in I_j} \frac{X-x_j}{x_i-x_j}\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-4929a1d023b7cc121a9c4e33135fd90b_l3.png "Rendered by QuickLaTeX.com")
have the properties
![\[e_i(x_i)=1\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-3e5e2e2331e68c99c6eacb75eaccadd8_l3.png "Rendered by QuickLaTeX.com")
![\[\bigwedge_{j \in I_i} e_i(x_j)=0\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-67922ef2d91d42ae714fdb33bae04534_l3.png "Rendered by QuickLaTeX.com")
or
![\[\bigwedge_{i \in I}\bigwedge_{j \in J} e_i(x_j)=\delta_{i,j}\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-d13d17824c5da13dd8cc5cb8422c6744_l3.png "Rendered by QuickLaTeX.com")
where 
is the Kronecker symbol, which is 0 if the two indices differ and 1 if they are equal.
Or as congruence:
![\[\bigwedge_{i \in I}\bigwedge_{j \in J} e_i(X)\equiv \delta_{i,j} \mod X-x_j\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-5b8a61f15f2b6236a85ec1e866440e55_l3.png "Rendered by QuickLaTeX.com")
Then we can just combine this and use
![\[p(X) =\sum_{i \in I} y_i e_i(X)\]](http://brodowsky.it-sky.net/wp-content/ql-cache/quicklatex.com-7887d71abe548841cf5ed08451d9d9a9_l3.png "Rendered by QuickLaTeX.com")
This can easily be written as a Ruby function
def fun_calc(pairs)
n = pairs.size
result = lambda do |x|
y = 0
n.times do |i|
p_i = pairs[i]
x_i = p_i[0].to_r
y_i = p_i[1].to_r
z = y_i
n.times do |j|
if (j != i)
p_j = pairs[j]
x_j = p_j[0]
z *= (x - x_j) / (x_i - x_j)
end
end
y += z
end
y
end
result
end
This takes a list of pairs as a parameter and returns the polynomial function als lambda.
It can be used like this:
lop = [[0, 0], [1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 64]]
f = fun_calc(lop)
20.times do |x|
y = f.call(x)
puts sprintf("%6d -> %6d", x, y)
end
Put this together into a ruby program and add some parsing for the list of pairs or change the program each time you use it and all these „difficult“ questions „that 99.9% fail to solve“ are not just easy, but actually soluble automatically.
This is interesting for more useful applications. I assume that there will always be situations where a function is needed that meets certain exact values a certain inputs and is an interpolation or extrapolation of this.
Please observe that there are other interesting and useful ways to approach this:
- Use a „best“ approximation from a set of functions, for example polynomials with a given maximum degree
- use cubic splines, which are cubic polynomials within each section between two neighboring input values such that at the input values the two adjacent functions have the same value (
![y_i]( "Rendered by QuickLaTeX.com")
, off course), the same first derivative and the same second derivative.
, off course), the same first derivative and the same second derivative.For highway and railroad construction other curves are used, because the splines are making an assumption on what is the 
-axis and what is the 
-axis, which does not make sense for transport facilities. They are using a curve called Clothoid.
Use Java, C, Perl, Scala, F# or the programming language of your choice to do this. You only need Closures, which are available in Java 8, F#, Scala, Perl, Ruby and any decent Lisp dialect. In Java 7 they can be done with an additional interface as anonymous inner classes. And for C it has been described in this blog how to do closures.