The Cracked Safe Combination

Zero valid combinations exist. A 4-digit palindrome (ABBA) necessarily repeats digits (positions 1=4 and 2=3), making it impossible for all four digits to be different. The constraints are mutually contradictory. If the all-different constraint is relaxed to A≠B, valid solutions include 1991, 2882, 3773, 4664 (all divisible by 11, digits summing to 20).

This is a trick question that tests whether solvers carefully check all constraints before computing. The palindrome and all-different-digits constraints directly conflict with each other for a 4-digit number. Step 1: A 4-digit palindrome has the form ABBA: position 1 = position 4, and position 2 = position 3. Step 2: Constraint: all four digits are different. But in ABBA, digit at position 1 = digit at position 4 (both equal A). This means positions 1 and 4 are the same digit — violating the all-different requirement immediately. Step 3: No matter what values A and B take, a 4-digit palindrome always repeats digits. It cannot have all four digits different. Step 4: Therefore, no 4-digit palindrome can have all four digits different. The two constraints are mutually exclusive. Step 5: What if the problem intended 'at least two different digits'? Then form ABBA with A+A+B+B = 2A+2B = 20 → A+B = 10, and divisible by 11. ABBA = 1001A + 110B = 11(91A + 10B). So every ABBA is divisible by 11! Then solutions with A+B=10 and A≠B: (1,9)→1991, (2,8)→2882, (3,7)→3773, (4,6)→4664, (9,1)→9119, etc. Step 6: The intended lesson: always verify that all constraints are simultaneously satisfiable before solving. Key Insight: The puzzle is deliberately constructed to test whether you catch the contradiction early. Checking constraint compatibility before diving into computation is a critical skill — especially in interview settings where trick questions are designed to see if you blindly proceed or pause to verify.