In the sum 8 + ?8 + 28 + 83 + 8? = 2?8, the same digit replaces each question mark. Which digit should ? represent so that the equation holds true?

Introduction / Context:
This question is a classic missing digit puzzle where a single unknown digit ?, used in several positions, must be determined so that an arithmetic equation balances. The expression contains both two digit and three digit numbers formed with ?, and the same digit appears in every question mark. Our goal is to translate this into an algebraic equation and solve it consistently.

Given Data / Assumptions:

• The expression is 8 + ?8 + 28 + 83 + 8? = 2?8.
• The symbol ? stands for the same digit (0–9) in every place where it appears.
• ?8 is a two digit number with tens digit = ? and ones digit = 8.
• 8? is a two digit number with tens digit 8 and ones digit = ?.
• 2?8 is a three digit number with hundreds digit 2, tens digit = ?, and ones digit 8.

Concept / Approach:
We introduce a variable, say x, for the unknown digit ?. Then we express each composite number in terms of x using place value: ?8 becomes 10x + 8, 8? becomes 80 + x, and 2?8 becomes 200 + 10x + 8. Substituting these expressions into the equation converts it into a standard linear equation in x, which we can solve using ordinary algebra. Finally, we check which value of x between 0 and 9 satisfies the equation and is also a valid single digit.

Step-by-Step Solution:
Step 1: Let ? = x, where x is a digit between 0 and 9. Step 2: Rewrite each composite number: ?8 = 10x + 8. 8? = 80 + x. 2?8 = 200 + 10x + 8. Step 3: Substitute into the equation: 8 + (10x + 8) + 28 + 83 + (80 + x) = 200 + 10x + 8. Step 4: Collect constant terms on the left: 8 + 8 + 28 + 83 + 80 = 207. Step 5: Collect x terms on the left: 10x + x = 11x. So left side = 11x + 207. Step 6: Simplify the right side: 200 + 10x + 8 = 208 + 10x. Step 7: Set both sides equal: 11x + 207 = 208 + 10x. Step 8: Subtract 10x from both sides: 11x - 10x + 207 = 208. This gives x + 207 = 208, so x = 1.

Verification / Alternative check:
Substitute x = 1 back into the original expression. Then ?8 = 18, 8? = 81, and 2?8 = 218. The sum of the left side is 8 + 18 + 28 + 83 + 81. Compute step by step: 8 + 18 = 26, 26 + 28 = 54, 54 + 83 = 137, 137 + 81 = 218. This equals the right side, 218, confirming that x = 1 satisfies the equation exactly.

Why Other Options Are Wrong:
If ? were 2, 3 or 4, then the equation would become a different linear equation and the sum on the left would not match the number formed on the right. A quick check reveals mismatches; for example, with x = 2 we would have 28 and 82 instead, leading to a left side different from 228. Therefore, only x = 1 yields a correct and consistent equality.

Common Pitfalls:
Some learners may misinterpret ?8 or 8? as multiplication expressions rather than two digit numbers. Another common mistake is to try each digit by guesswork rather than setting up an equation, which is slower and more error prone. Working with place values and forming a clean algebraic equation provides a reliable method that scales to more complex puzzles.

Final Answer:
The digit represented by ? that makes the equation true is 1.