2010 AMC 10B Problems/Problem 5

Problem

A month with days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?

Solution

In this month there are four weeks and three remaining days. Any 7 days must have exactly one Monday and one Wednesday, so it works if the last days have the same number of Mondays and Wednesdays. We have three choices: Monday, Tuesday, Wednesday; Thursday, Friday, Saturday; Friday, Saturday, Sunday. The number of days the month can start on are Monday, Thursday, and Friday, for a final answer of

Solution 2

Let's make a calendar to visualize the situation better.

\begin{table}[] \begin{tabular}{lllllll} \hline \multicolumn{1}{|l|}{1} & \multicolumn{1}{l|}{2} & \multicolumn{1}{l|}{3} & \multicolumn{1}{l|}{4} & \multicolumn{1}{l|}{5} & \multicolumn{1}{l|}{6} & \multicolumn{1}{l|}{7} \\ \hline \multicolumn{1}{|l|}{8} & \multicolumn{1}{l|}{9} & \multicolumn{1}{l|}{10} & \multicolumn{1}{l|}{11} & \multicolumn{1}{l|}{12} & \multicolumn{1}{l|}{13} & \multicolumn{1}{l|}{14} \\ \hline \multicolumn{1}{|l|}{15} & \multicolumn{1}{l|}{16} & \multicolumn{1}{l|}{17} & \multicolumn{1}{l|}{18} & \multicolumn{1}{l|}{19} & \multicolumn{1}{l|}{20} & \multicolumn{1}{l|}{21} \\ \hline \multicolumn{1}{|l|}{22} & \multicolumn{1}{l|}{23} & \multicolumn{1}{l|}{24} & \multicolumn{1}{l|}{25} & \multicolumn{1}{l|}{26} & \multicolumn{1}{l|}{27} & \multicolumn{1}{l|}{28} \\ \hline 29 & 30 & 31 & & & & \end{tabular} \end{table}

See Also

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