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A company has $8$ employees $p_1,p_2,p_3,p_4,p_5,p_6,p_7$ and $p_8$. Every weekend must be employed at least $3$ (or more) of them. These employees have the following restrictions:

  • Employees $p_1,p_2$ and $p_3$ must work every other weekend.
  • Employees $p_4,p_5,p_6,p_7$ and $p_8$ must work one weekend in and two weekends out (one every three).

I wonder if this problem has solution.

popi
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    No, but if you shift on of the "every other weekend" workers to "one weekend on, two off" and still pay the higher salary then it is possible – Henry Mar 06 '18 at 17:54
  • I suspected the same. But, Can you explain why no? – popi Mar 06 '18 at 18:12
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    It is also possible if you interpret "one weekend in and two weekends out" to allow only require that they work 1/3 of the weekends overall, not the more rigid requirement that they work every third weekend. – Paul Sinclair Mar 07 '18 at 01:19
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    As for why the strict interpretation is not possible, there are only two possibilities (after a relabeling). $p_1, p_2$ always work together. Either $p_3$ works with them (in which case there are only 2 workers available on the 4th weekend), or $p_3$ works the opposite weekend, in which case there is only one other worker available to work with $p_3$ on the 6th weekend. – Paul Sinclair Mar 07 '18 at 01:27
  • Have you considered the possibility that 3 OR MORE employees have to work each weekend? – popi Mar 07 '18 at 11:09

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