For a given list of three numbers, the operation “changesum” replaces each number in the list with the sum of the other two. For example, applying “changesum” to 3, 11, 7 gives 18, 10, 14. Arav starts with the list 20, 2, 3 and applies the operation “changesum” 2023 times. What is the largest difference between two of the three numbers in his final list?
A. 17
B. 18
C. 20
D. 2021
E. 2023
B. 18
Let the three numbers in the list be X, Y and Z, where we will assume that X ≥ Y ≥ Z. The differences then can be written as X − Y, X − Z and Y − Z. After the operation “changesum” has been applied to the list, the values in the new list become Y + Z, X + Z and X + Y. The differences between these new values are (X + Z) − (Y + Z), (X + Y) − (Y + Z) and (X + Y) − (X + Z) which are equal to X − Y, X − Z and Y − Z. Therefore it can be seen that the differences between the numbers in the list after applying the operation “changesum” are the same as the differences between the numbers in the list before applying the operation “changesum”. Hence, the largest difference between two numbers in Arav’s list after applying “changesum” 2023 times will be equal to the largest difference between two numbers in the original list, that is 20 − 2, which is equal to 18.
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