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Puzzling

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    $\begingroup$ 25 > 24, but the following 25-day schedule has no consecutive sum of 11: 2 1 1 12 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 $\endgroup$ Commented Aug 15, 2024 at 22:19
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    $\begingroup$ @RobPratt - Ah, I see where I went wrong; it's possible for two numbers in the set to correspond to the same number which is not in the set, so splitting it into 24 distinct pairs of numbers isn't correct. $\endgroup$ Commented Aug 15, 2024 at 22:59
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    $\begingroup$ @RobPratt - I've fixed the logic. With this correction, I believe the minimum is actually 27 days. $\endgroup$ Commented Aug 15, 2024 at 23:11
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    $\begingroup$ Yes, with 48 pills, a 26-day 11-free schedule is possible, but a 27-day 11-free schedule is not. $\endgroup$ Commented Aug 16, 2024 at 1:30
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    $\begingroup$ I cleaned up my answer a bit to make it clear which part of it was originally incorrect (since it's now the accepted answer). I don't think my answer should have been accepted, as Ankoganit's answer was posted before mine and also supplies a sufficient proof. $\endgroup$ Commented Aug 17, 2024 at 5:39