Logic puzzle: find negation of three logic variables using exactly two negations and any number of conjunctions and disjunctions We're given a programming environment in which it is possible to introduce 
new logical variables and issue logical commands with operations AND, OR and NOT,
such as A = (X AND Y) OR Z.
However, TRUE and FALSE are not defined, so that it is NOT allowed to write A = X OR TRUE.
At the beginning, we are given three logical variables X, Y and Z, initialized to arbitrary values.
We need to make a list of logical commands that in total contains AT MOST TWO operations NOT,
so that after the commands are executed the variables X, Y and Z contain their respective negations.
I have tried several times to solve this using these ideas:


*

*Some random logic expressions based on common negation of three given variables

*Try to find out when the value of some negation is true based on number of true and false values.


None of these worked for me...
NOTE: I'm not sure if this task is solvable, but even if it has no solution, that has to be proved as well
 A: The procedure is on the left, and on the right is written a formula equivalent to the one on the left.
\begin{array}{rcl|cl}
A_{xy}&=& X\land Y \\
A_{xz}&=& X\land Z \\
A_{yz}&=& Y\land Z \\
A_{xyz}&=& X\land Y\land Z \\
B&=&A_{xy}\lor A_{xy}\lor A_{yz} & \equiv& (X\land Y)\lor (X\land Z)\lor (Y\land Z)\\
C&=&\lnot B & \equiv& (\lnot X\lor\lnot Y)\land(\lnot X\lor\lnot Z)\land(\lnot Y\lor\lnot Z)\\
D_x&=& X\land C & \equiv& X\land\lnot Y\land\lnot Z\\
D_y&=& Y\land C & \equiv& \lnot X\land Y\land\lnot Z\\
D_z&=& Z\land C & \equiv& \lnot X\land\lnot Y\land Z\\
E&=&A_{xyz}\lor D_x\lor D_y\lor D_z & \equiv& (X\land Y\land Z)\lor (X\land\lnot Y\land\lnot Z)\ \lor\\&&&& (\lnot X\land Y\land\lnot Z) \lor (\lnot X\land\lnot Y\land Z)\\
F&=&\neg E & \equiv& (\lnot X\lor \lnot Y\lor \lnot Z)\land (\lnot X\lor Y\lor Z)\ \land\\&&&& ( X\lor\lnot  Y\lor Z) \land ( X\lor Y\lor\lnot  Z)\\
G_{xy} &=& A_{xy}\land F &\equiv & X\land Y\land \lnot Z\\
G_{xz} &=& A_{xz}\land F &\equiv & X\land\lnot Y\land  Z\\
G_{yz} &=& A_{yz}\land F &\equiv & \lnot X\land Y\land Z\\
H&=& C\land F &\equiv & \lnot X\land \lnot Y\land \lnot Z\\
I_x&=&D_y\lor D_z\lor G_{yz}\lor H&\equiv &\lnot X\\
I_y&=&D_x\lor D_z\lor G_{xz}\lor H&\equiv &\lnot Y\\
I_z&=&D_x\lor D_y\lor G_{xy}\lor H&\equiv &\lnot Z\\
\end{array}
