Rajitha has a calculator on which the digit keys have been interchanged in three disjoint pairs, and from the given correct outputs she deduces which digit keys are unchanged. Which set of three digit keys remains in its original position?

Introduction / Context:
This is a logical puzzle based on a faulty calculator whose digit keys have been swapped in pairs. The puzzle provides a few sample calculations where the inputs are sequences of keys pressed and the outputs are numerically correct. From this information, we must infer which digit keys were never moved and therefore still produce their original digits. Such questions test pattern recognition, logical reasoning, and comfort with number operations.

Given Data / Assumptions:
- The calculator has digit keys 0 through 9.- Exactly three disjoint pairs of digit keys have swapped positions.- When Rajitha presses a key labelled d, the calculator internally reads some other digit f(d) because of the swap.- The calculator still performs arithmetic correctly on the internal digits and displays correct numerical results.- The following keypresses and displayed outputs are known: 1. Pressed 95 + 84, displayed 128. 2. Pressed 128 + 64, displayed 195. 3. Pressed 912 + 43, displayed 942. 4. Pressed 178 + 19, displayed 192.

Concept / Approach:
Let f(d) denote the actual digit value read when the key labelled d is pressed. Because only pairs were swapped, f is a permutation made up of three transpositions and four fixed points. For each equation, the sum of the numbers built from f applied to the pressed digits must equal the displayed result. By writing each pressed number digit by digit in terms of f and forming equations, we can deduce the mapping. Once the mapping is known, the fixed points give us the digits that were not moved.

Step-by-Step Solution:
- From 95 + 84 = 128, form 10 * f(9) + f(5) plus 10 * f(8) + f(4) equals 128.- From 128 + 64 = 195, form 100 * f(1) + 10 * f(2) + f(8) plus 10 * f(6) + f(4) equals 195.- From 912 + 43 = 942, form 100 * f(9) + 10 * f(1) + f(2) plus 10 * f(4) + f(3) equals 942.- From 178 + 19 = 192, form 100 * f(1) + 10 * f(7) + f(8) plus 10 * f(1) + f(9) equals 192.- Solving this system carefully, while remembering that each digit 0 through 9 appears exactly once as a value of f(d), yields the unique mapping: f(2) = 4, f(4) = 2, f(3) = 8, f(8) = 3, f(5) = 6, f(6) = 5.- All other digits are fixed: f(0) = 0, f(1) = 1, f(7) = 7, f(9) = 9.- Therefore, the digits 0, 1, 7, and 9 are unchanged, and any set containing only these digits is a correct set of unchanged keys.- Among the answer choices, the triple 7, 0, 1 consists entirely of unchanged digits.

Verification / Alternative check:
Check one of the equations using the mapping. For 95 + 84, we have f(9) = 9 and f(5) = 6, so the first number is 96. Also f(8) = 3 and f(4) = 2, so the second number is 32. Their sum is 96 + 32 = 128, which matches the displayed result. Similar checks on the other three sums confirm the mapping is consistent. This validates the conclusion about which digits are unchanged.

Why Other Options Are Wrong:
- 9, 0, 3: Digit 3 is involved in a swap with 8, so it is not unchanged.- 1, 3, 8: Both 3 and 8 are swapped, so this set cannot represent unchanged keys.- 2, 4, 1: Digits 2 and 4 are swapped with each other, so they are not fixed.- 5, 6, 9: Digits 5 and 6 form another swapped pair and are not unchanged.

Common Pitfalls:
Learners may try to guess the answer from only one equation or misunderstand that the keys were swapped in pairs, not arbitrarily. Ignoring the permutation structure leads to contradictions. A systematic approach that uses all the given equations and the condition that each digit is used exactly once resolves the puzzle uniquely. Always keep track of which digits have already been assigned in the mapping to avoid conflicts.

Final Answer:
The set of digit keys that remain in their original positions is 7, 0, 1.