You have the row:
$$ 4,\ 2,\ 6,\ 1,\ 5,\ 3. $$
The only operation allowed is to exchange two adjacent cards whose sum is odd.
Can you reach:
$$ 1,\ 2,\ 3,\ 4,\ 5,\ 6? $$
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The impossible row of cards
Hints
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- Both relative orders change, which contradicts the invariant.
- You can only exchange adjacent cards with an odd sum, that is, an even card with an odd card.
- This implies that: the relative order of the even cards to each other 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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