2022 AMC 10B Problem 1

The symbol $\diamond$ just means "take the distance between the two numbers": $x \diamond y = |x - y|$. The brackets tell you which pair to work out first, so do the innermost bracket first in each half.

1. Left half, $1 \diamond (2 \diamond 3)$. Inside first: $2 \diamond 3 = |2 - 3| = 1$. Then $1 \diamond 1 = |1 - 1| = 0$.
2. Right half, $(1 \diamond 2) \diamond 3$. Inside first: $1 \diamond 2 = |1 - 2| = 1$. Then $1 \diamond 3 = |1 - 3| = 2$.
3. Subtract. $0 - 2 = -2$.

The answer is $\boxed{\textbf{(A)}\ -2}$.

With ordinary addition, $(a + b) + c$ and $a + (b + c)$ are always equal, so it is tempting to think brackets never matter. For a made-up operation like "distance between", they can matter a lot, and this problem is built to show that. When a question invents a new operation, follow the definition step by step and do not borrow rules from ordinary arithmetic.

Work from the inside out.

First half: $2\diamond 3 = |2-3| = 1$, then $1\diamond 1 = |1-1| = 0$.

Second half: $1\diamond 2 = |1-2| = 1$, then $1\diamond 3 = |1-3| = 2$.

The difference is $0 - 2 = -2$.

The answer is $\boxed{\textbf{(A)}\ -2}$.