You start with the row of cards 4, 2, 6, 1, 5, 3. The only operation allowed is to exchange two adjacent cards whose sum is odd.
Can this be achieved 1, 2, 3, 4, 5, 6?
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The row impossible of cards
The rule seems local and modest, but it imposes a profound limitation on the entire order. It is a good example of how a small ban is enough to close many doors.
Hints
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- Not all adjacent transpositions are allowed.
- You can only exchange cards of different parity.
- Ask yourself what relative order never changes.
Solution
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Answer: No, it is impossible. You can only exchange adjacent cards with an odd sum, that is, an even card with an odd card. That implies that: - the relative order of the even cards to each other never changes,
- and the relative order of the odd ones to each other also never changes. In the initial row: - pairs: $4,2,6$,
- odd: $1,5,3$. In the target row: - pairs: $2,4,6$,
- odd: $1,3,5$. Both relative orders change, which contradicts the invariant.
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