In the cryptarithm 8A9 − 6B2 + 4C6 = 723, A, B, and C each represent a single digit from 0 to 9. What is the maximum possible value that digit B can take so that the equation still holds true?

Introduction:
This question is a simple cryptarithm, where letters stand for decimal digits. You are given an arithmetic expression involving three unknown digits A, B, and C, and you must determine the maximum possible digit B that satisfies the equation. This tests logical reasoning, digit manipulation, and equation solving skills.

Given Data / Assumptions:

• The expression is 8A9 − 6B2 + 4C6 = 723.
• A, B, and C are digits from 0 to 9.
• 8A9 means the three-digit number 800 + 10A + 9, and similarly for 6B2 and 4C6.
• We must find the greatest possible value of B that makes the equation true with some digits A and C.

Concept / Approach:
We first rewrite the expression by expanding each three digit form. Then we simplify to link A, B, and C in a linear equation. Since the equation must hold for digits, we then search for integer digit values that satisfy this relationship and identify the maximum B that works.

Step-by-Step Solution:
Step 1: Rewrite each number: 8A9 = 800 + 10A + 9, 6B2 = 600 + 10B + 2, 4C6 = 400 + 10C + 6. Step 2: Substitute into the equation: (800 + 10A + 9) − (600 + 10B + 2) + (400 + 10C + 6) = 723. Step 3: Combine constants: 800 + 9 + 400 + 6 − 600 − 2 = 613. So we get: 613 + 10A − 10B + 10C = 723. Step 4: Simplify further: 10(A − B + C) = 723 − 613 = 110. Step 5: Divide by 10: A − B + C = 11. Step 6: A, B, C are digits from 0 to 9. We want the maximum B such that A − B + C = 11 is possible. The maximum B occurs when A and C are as large as possible, that is A = 9 and C = 9. Step 7: With A = 9 and C = 9, we get 9 − B + 9 = 11, so 18 − B = 11 which gives B = 7.

Verification / Alternative check:
Substitute A = 9, B = 7, C = 9 into the original expression: 8A9 = 899, 6B2 = 672, 4C6 = 496. Then 899 − 672 + 496 = 227 + 496 = 723, which matches the required result. Therefore B = 7 works and is valid.

Why Other Options Are Wrong:

• 4, 1, 3, and 5 are smaller than 7 and do not represent the maximum possible value for B.
• For larger digits like 8 or 9, the equation A − B + C = 11 cannot be satisfied with digits A and C in the range 0 to 9.

Common Pitfalls:
Some students try random substitutions without first simplifying the equation, which is inefficient and error prone. Others forget that A, B, and C must be digits and might mistakenly consider values outside 0 to 9. Always convert the cryptarithm into a clean algebraic equation and then reason about digit constraints.

Final Answer:
The maximum possible value of digit B is 7.