3D Treasure Hunt Vihaan Goyal

I am using a trick called the manhattan distance to solve those question. The manhattan distance states in a 3d plane D=|x-x_t|+|y-y_t|+|z-z_t|. Using this I will find the x coordinates in terms of d terms. First guess(0,0,0)

(0,0,0) s1

D1=x_t+y_t+z_t

(N,0,0)

X_t is less  than or equal to n so |n-x_t| is n-x_t

D2=(n-x_t)+y_t+z_t

D1-d2=2x_t-n. X_t=((D1-D2)+n)/2

(0,n,0)

D3=x_t+(n-y_t)+z_t

D1-D3=2y_t-n y_t=((D1-D3+N))/2

So z=D1-x_t-y_t

Answer is (D1-D2+N/2,D1-D3+N/2,D1-X_t- y_t)