Equality in Stages
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are written on the board, and David is supposed to change them in stages. At each stage, David is allowed to choose two numbers and change them by 1, that is, to add 1 to both, subtract 1 from both, or add 1 to one and subtract 1 from the other.
Can David, after a number of stages, reach a situation where all the numbers on the board are equal? If so, show an example, and if not, explain your answer in detail.
Difficulty level (1 very easy - 10 very hard): 3
Topics:
Combinatorics
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Invariants
Logic
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Reasoning / Logic
Number Theory
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Division
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Parity (Even/Odd)
Combinatorics
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Case Analysis / Checking Cases
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Processes / Procedures
- Grossman Math Olympiad, 2017, Juniors Question 3
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