Author: Belukhov $H$.

In a certain state, addition and subtraction are denoted by the signs "!" and "?", but you do not know which sign corresponds to which operation. Each operation is applied to two numbers, but you do not know whether subtraction involves subtracting the left number from the right or the right from the left. For example, the expression \$a?b\$ represents one of the following: \$a - b, b - a\$ or \$a + b\$. You do not know how numbers are written in this state, but variables \$a, b\$ and parentheses are used as usual. Explain how to use them and the signs "!", "?" to write an expression that is guaranteed to equal $\$ 20 \mathrm{a}-18 \mathrm{~b} \$.
Note that both operations are sums, with only one of the addends possibly being multiplied by -1. Therefore, when calculating, we will not swap the addends. There are four possible combinations of operations. For example, the variant "?L" means that when performing "?", the Left addend is multiplied by -1.

Consider the value of the expression \$x?(y!y)!y?y!y\$ (operations are performed from left to right).

?L: $\$-(-x+2 y+y)+y+y=x-y . \$$

?P: $\$ x-2 y+y-y+y=x-y . \$$

!L: $\$-(-(x+0)+y+y)+y=x-y \cdot \$$

$!\Pi: \$ x+0-y+y-y=x-y . \$$

Thus, we can compute the ordinary difference. Then we can also compute the sum, since $\$ x+y=x-(x-$ $\mathrm{x}$ - y)\$. Consequently, we can obtain any linear combination of \$ $\$ \mathrm{a}$ and $\$ \mathrm{~b} \$ \mathrm{c}$ with integer coefficients.