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In a year, there are more Mondays than Fridays, and more Sundays than Wednesdays. Which day of the week is the first day in March of that year?

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  • $\begingroup$ How is this possible. As $365\equiv 1\pmod 7$, the day appears as the first of the year is only $53$ times that year. Other days are all $52$ times.One can consider a leap year, as $366\equiv 2\pmod 7$, but, there are two consecutive days are $53$ times. $\endgroup$ – user249332 Sep 20 '15 at 10:06
  • $\begingroup$ @SubhadeepDey It must be a leap year that starts on Sunday. $\endgroup$ – N. F. Taussig Sep 20 '15 at 11:24
  • $\begingroup$ @N.F.Taussig, oh yes, I missed that. Thank you. $\endgroup$ – user249332 Sep 20 '15 at 17:28
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It is definitely a leap year and the first day is Sunday. Because, if we assume that it is not a leap year (365 days), then we have just one day with one more occurrence compared to other days. however, we need two days with higher occurrence. Now we can write week days with the order:

Sun-Mon-Tue-Wed-Thu-Fri-Sat

By repeating this sequence 52 times, we get 364 days. So, the remaining two days are Sunday and Monday which have one more occurrence than the others. If the sequence changes in any possible way, one of the two days (Sunday and Monday) or both would not meet the conditions that are mentioned.

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