### Instruction: For real numbers $x$ and $y,$ find all possible values of
\[\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor.\]Enter all possible values, separated by commas.### Response: I recognize that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x,$ so it rounds $x$ down to the nearest integer.
I also know that $\lfloor x + y \rfloor$ is not necessarily equal to $\lfloor x \rfloor + \lfloor y \rfloor,$ because the sum of two numbers might round down differently than each number individually.
For example, $\lfloor 1.5 + 1.5 \rfloor = 3,$ but $\lfloor 1.5 \rfloor + \lfloor 1.5 \rfloor = 2.$
So, I want to see how the difference $\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor$ depends on the fractional parts of $x$ and $y.$
I can write $x = \lfloor x \rfloor + \{x\}$ and $y = \lfloor y \rfloor + \{y\},$ where $\{x\}$ and $\{y\}$ are the fractional parts of $x$ and $y.$
Then, $\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor = \lfloor \lfloor x \rfloor + \{x\} + \lfloor y \rfloor + \{y\} \rfloor - \lfloor x \rfloor - \lfloor y \rfloor.$
I can simplify this by canceling out the integer parts: $\lfloor \lfloor x \rfloor + \{x\} + \lfloor y \rfloor + \{y\} \rfloor - \lfloor x \rfloor - \lfloor y \rfloor = \lfloor \{x\} + \{y\} \rfloor.$
Now, I only need to consider the fractional parts of $x$ and $y.$
I know that $\{x\}$ and $\{y\}$ are both in the interval $[0,1),$ meaning they are greater than or equal to zero, but less than one.
Therefore, their sum $\{x\} + \{y\}$ is also in the interval $[0,2),$ meaning it is greater than or equal to zero, but less than two.
The possible values of $\lfloor \{x\} + \{y\} \rfloor$ depend on whether $\{x\} + \{y\}$ is less than one, or greater than or equal to one.
If $\{x\} + \{y\} < 1,$ then $\lfloor \{x\} + \{y\} \rfloor = 0.$
If $\{x\} + \{y\} \geq 1,$ then $\lfloor \{x\} + \{y\} \rfloor = 1.$
Therefore, the only possible values of $\lfloor x + y \rfloor - \lfloor x \rfloor - \lfloor y \rfloor$ are 0 and 1.