"Continue the pattern" and the unprovability of the existence of a single solution Recently (half an hour ago) an associate of mine asked me to prove that:
Given $x=1\implies y=2, x=2\implies y=4, x=3\implies y=6, x=4 \implies y=a$, prove that there is no single solution to a.
With my current mathematical knowledge, I was stumped. After a few minutes of attempts, I asked him for his solution. He gleefully gave me the "arbitrary function" explanation, and while that was good enough for an intuitive explanation, I wasn't satisfied. I pointed out that that was not a valid proof to him, and a little back and forth ensued.
After a while, I began to think: how do I prove that? But of course, I cannot do it on my own. So, I have come to ask ye of MathExchange for a proof.
My current insights are:

*

*It cannot be proven that there are any, one, or many solutions (this requires proof as well).

 A: Passing a line through 2 given Points $(1,A),(2,B)$ is always Possible. Passing a line through 3 given Points $(1,A),(2,B),(3,C)$ may not always be Possible , unless the Points are co-linear. It is always Possible to put it on a quadratic curve.
Like-wise it is not always Possible to put a quadratic curve on 4 given Points $(1,A),(2,B),(3,C),(4,D)$ , though we can always put a cubic curve.
In other words , given $N$ Points like the earlier Cases , we can always put those Points on a curve of Degree $N-1$ , or higher.
Now , we are given the Points $(1,2),(2,4),(3,6),(4,a)$ , but what-ever we choose for $a$ , we can put the Points on a curve of Degree 3 or higher.
$(1,2),(2,4),(3,6),(4,1)$ : $-(7 x^3)/6 + 7 x^2 - (65 x)/6 + 7$

$(1,2),(2,4),(3,6),(4,10)$ : $x^3/3 - 2 x^2 + (17 x)/3 - 2$

$(1,2),(2,4),(3,6),(4,-10)$ : $-3 x^3 + 18 x^2 - 31 x + 18$

$(1,2),(2,4),(3,6),(4,0)$ : $-(4 x^3)/3 + 8 x^2 - (38 x)/3 + 8$

$(1,2),(2,4),(3,6),(4,-1)$ : $-(3 x^3)/2 + 9 x^2 - (29 x)/2 + 9$

ALL IMAGES AND CALCULATIONS CURTESY OF WOLFRAM ALPHA
Hence , no matter what $a$ is , we can get a function for that. Hence $a$ is not unique.
It will be unique (or limited in range) if we puts Consistent Constraints like (1) We want a Degree 2 curve. (2) We want a linear curve. Etc.
