Belukhov H.

In a certain state, addition and subtraction are denoted by the signs "!" and "?", but it is unknown to you which sign corresponds to which operation. Each operation is applied to two numbers, but it is unknown to you 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 $a+b$. It is unknown to you 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 "!" and "?" to write an expression that is guaranteed to equal $20a-18b$.
First, note that the expression ( $a$ ? $a$ ) ! ( $a$ ? $a$ ) is always equal to zero. In the future, we can use 0, implying that this exact expression should be written instead.

The expression ( $x$ ? 0) ? (0 ? y) is always equal to $x+y$. Similarly, we can now use the + operation with two arguments.

Finally, the expression 0 ? ((0 ! ( $x$ ! 0)) ? 0) is always equal to $-x$.

Now it is easy to write the desired expression:

![](https://cdn.mathpix.com/cropped/2024_05_06_dbba6af20ebd45cbbb54g-17.jpg?height=202&width=1118&top_left_y=2190&top_left_x=467)

The hexagons contain digits and arithmetic operation signs as shown in the figure. The task is to start from one of the hexagons and move to an adjacent one, visiting each one exactly once. As you do this, you need to write down what is written in them in a line, and in the end, obtain a correct equation. Which one?

## Answer

$5=4: 2+3$.

The given answer is the only one.