Shamir Secret Sharing - How to reconstruction secret? Example I try implement Shamir Secret Sharing.

In this video: https://www.youtube.com/watch?v=j7Ngtl1cCpY is example.



But for another three share holders is can't reconstruction secrete.

For example:
a)
a(1) = 15
a(2) = 6
a(4) = 10


b)
a(1) = 15
a(2) = 6
a(5) = 6
But for another three share holders is can't reconstruction secrete.


Sure they can.

In the field $GF(17)$, we have:

$$\frac{43}{3} = \frac{9}{3} = 3$$

because $43 = 9 \pmod {17}$, and $3$ is the value that, when multiplied by 3, gives us 9 modulo 17.

And, we have:

$$\frac{23}{2} = \frac{6}{2} = 3$$

beause $23 = 6 \pmod {17}$, and $3$ is the value that, when multipled by 2, gives us 9 modulo 17.

Remember that, when we work in a finite field, division isn't exactly the same as it is when we're working with in $\mathbb{R}$; for example $\frac{1}{2}$ is not 0.5 (which is meaningless in $GF(17)$, but instead it is 9.