Rule 1: Only + and - operators may be used.
Rule 2: Only one operation per step.
Rule 3: The - operator must return the result into the operand to the left of the operator (e.g. Z - X must be put into Z, not into X)
Rule 4: All operations are integer operations (5 / 3 = 1, not rounded but truncated to 1).

X and Y can be any integer number (and must work for any number) at step 0. Z = 0 at step 0.

In four steps, swap the values in X and Y.

Rule 1: Only the + operator may be used.
Rule 2: Only one operation per step.

Initial values: X = 10, Y = 15, Z = 20.
After 3 steps, make Y = 55 and Z = 60 (X can be anything).

Rule 1: Only + - / and * operators may be used.
Rule 2: Only one operation per step.
Rule 3: All operations are integer operations (5 / 3 = 1, not rounded but truncated to 1).
Rule 4: The / and - operators must return the result into the operand to the left of the operator (e.g. Z - X must be put into Z, not into X).

X at step 0 is 8 when Y at step 0 is 9, and vice versa. The algorithm must work for both (X=8, Y= 9) and (X=9, Y=8). Z is empty. After two steps, Z must equal 9 (for both initial sets of values).

Rule 1: Only + and - operators may be used.
Rule 2: Only one operation per step.
Rule 3: All operations are integer operations (5 / 3 = 1, not rounded but truncated to 1).
Rule 4: The - operator must return the result into the operand to the left of the operator (e.g. Z - X must be put into Z, not into X).

For any initial values of X, Y, and Z, after four steps Z must equal X + 2Y - 3Z (the final value of Z must equal the initial value of X + 2x the initial value of Y - 3x the initial value of Z).

Example inputs and output: X=1, Y=2, Z=3. Output: Z=-4
(1 + 2(2) - 3(3))

X = X + Y
Z = Z - Y
Y = X + Z
X = X - Y

Result X < Y

               X =  5, Y = 7, Z =  0
X = 5 + 7      X = 12, Y = 7, Z =  0
Z = 0 - 7      X = 12, Y = 7, Z = -7
Y = 12 + -7    X = 12, Y = 5, Z = -7
X = 12 - 5     X =  7, Y = 5, Z = -7

Result X = Y

               X = 2, Y = 2, Z =  0
X = 2 + 2      X = 4, Y = 2, Z =  0
Z = 0 - 2      X = 4, Y = 2, Z = -2
Y = 4 + -2     X = 4, Y = 2, Z = -2
X = 4 - 2      X = 2, Y = 2, Z = -2

Result X > Y

               X = 3, Y = 1, Z =  0
X = 3 + 1      X = 4, Y = 1, Z =  0
Z = 0 - 1      X = 4, Y = 1, Z = -1
Y = 4 + -1     X = 4, Y = 3, Z = -1
X = 4 - 3      X = 1, Y = 3, Z = -1

X = Z + Z
Y = X + Y
Z = X + Z

Result

               X = 10, Y = 15, Z = 20
X = 20 + 20    X = 40, Y = 15, Z = 20
Y = 40 + 15    X = 40, Y = 55, Z = 20
Z = 40 + 20    X = 40, Y = 55, Z = 60

Z = X / Y
Z = Y + Z

Result X < Y

               X = 8, Y = 9, Z = 0
Z = 8 / 9      X = 8, Y = 9, Z = 0
Z = 9 + 0      X = 8, Y = 9, Z = 9

Result X > Y

               X = 9, Y = 8, Z = 0
Z = 9 / 8      X = 9, Y = 8, Z = 1
Z = 8 + 1      X = 9, Y = 8, Z = 9

Y = Y - Z
Y = Y + Y
X = X - Z
Z = X + Y

Result X < Y < Z

               X =  1, Y =  2, Z = 3 (1 + 2*2 - 3*3 = -4)
Y = 2 - 3      X =  1, Y = -1, Z = 3
Y = -1 + -1    X =  1, Y = -2, Z = 3
X = 1 - 3      X = -2, Y = -2, Z = 3
Z = -2 + -2    X = -2, Y = -2, Z = -4

Result X < Z < Y

               X =  1, Y =  3, Z = 2 (1 + 2*3 - 3*2 = 1)
Y = 3 - 2      X =  1, Y =  1, Z = 3
Y = 1 + 1      X =  1, Y =  2, Z = 3
X = 1 - 2      X = -1, Y =  2, Z = 3
Z = -1 + 2     X = -1, Y =  2, Z = 1

Result Y < X < Z

               X =  2, Y =  1, Z =  3 (2 + 2*1 - 3*3 = -5)
Y = 1 - 3      X =  2, Y = -2, Z =  3
Y = -2 + -2    X =  2, Y = -4, Z =  3
X = 2 - 3      X = -1, Y = -4, Z =  3
Z = -1 + 2     X = -1, Y = -4, Z = -5

Result Y < Z < X

               X = 3, Y =  1, Z =  2 (3 + 2*1 - 3*2 = -1)
Y = 1 - 2      X = 3, Y = -1, Z =  2
Y = -1 + -1    X = 3, Y = -2, Z =  2
X = 3 - 2      X = 1, Y = -2, Z =  2
Z = 1 + -2     X = 1, Y = -2, Z = -1

Result Z < X < Y

               X = 2, Y = 3, Z = 1 (2 + 2*3 - 3*1 = 5)
Y = 3 - 1      X = 2, Y = 2, Z = 1
Y = 2 + 2      X = 2, Y = 4, Z = 1
X = 2 - 1      X = 1, Y = 4, Z = 1
Z = 1 + 4      X = 1, Y = 4, Z = 5

Result Z < Y < X

               X = 3, Y = 2, Z = 1 (3 + 2*2 - 3*1 = 4)
Y = 2 - 1      X = 3, Y = 1, Z = 1
Y = 1 + 1      X = 3, Y = 2, Z = 1
X = 3 - 1      X = 2, Y = 2, Z = 1
Z = 2 + 2      X = 2, Y = 2, Z = 4

Result X = Y = Z

               X = 1, Y = 1, Z = 1 (1 + 2*1 - 3*1 = 0)
Y = 1 - 1      X = 1, Y = 0, Z = 1
Y = 0 + 0      X = 1, Y = 0, Z = 1
X = 1 - 1      X = 0, Y = 0, Z = 1
Z = 0 + 0      X = 0, Y = 0, Z = 0

I'm too lazy to write out the other cases. It works.

After the nth operation...
0th: x = a, y = b, z = 0
1st:
2nd:
3rd: x = b or y = a (since you only have one op left to use)
4th: x = b and y = a

We can determine all the states for the first operation by exhaustion then eliminating those that either break a rule, result in losing a or b completely or are the exact duplicate of one we've written down previously. Since x and y are interchangeable at this point we only need to determine the way we could change one of them. You should get these possibilities for "1st":
x = x + x = 2a
x = x + y = a + b
x = x - y = a - b
z = x + x = 2a
z = x + y = a + b
z = x + z = a
z = y + y = 2b
z = z + y = b
z = z - y = a - b
Pick whichever looks best for solving this problem, use an op that moves it to the 3rd state, see if any can move it to the 4th.

x = x + y
z = z - y
y = x + z
x = x - y

x = z + z
y = y + x
z = z + x

z = x * y
z = z / 8

y = y - z
y = y + y
y = y - z
z = x + y

Z - Y
Z = Z + X
Y = Y + Z
X - Z

X = Z + Z
Y = Y + X
Z = Z + X

Z = X / Y
Z = Z + Y

Y = Y - Z
Y = Y + Y
X = X - Z
Z = Y + X