Number Pattern Problem: Matching First and Last Digits in Three-Digit Numbers

To determine the correct number, check the digits of each option to see if the hundreds digit equals the units digit:

• Option 1: $644$ - hundreds digit is $6$, units digit is $4$ (they are not equal).
• Option 2: $858$ - hundreds digit is $8$, units digit is $8$ (they are equal).
• Option 3: $677$ - hundreds digit is $6$, units digit is $7$ (they are not equal).
• Option 4: $550$ - hundreds digit is $5$, units digit is $0$ (they are not equal).

Therefore, the number that meets the condition where the hundreds digit equals the units digit is $\textbf{858}$.