.Problema para ajudar na escola: Quantas quintas?

Problema
(A partir do 6º ano do E. F. – Nível de dificuldade: Médio)

(ONEM, 2010) Em um determinado ano, um mês específico teve cinco sextas-feiras e cinco domingos. Quantas quintas-feiras teve o mês em questão?

Solução

• Vamos supor inicialmente que o primeiro domingo do mês em questão tivesse ocorrido na primeira semana do mês. Como ocorreram cinco domingos no mês, a distribuição mínima dos dias da semana nesse mês seria a mostrada na tabela abaixo.

[tex]\begin{array}{r|r|r|r|r|r|r|r}
&\text{domingo} &\text{segunda} & \text{ terça }\,&\text{quarta} &\text{quinta} &\text{ sexta }&\text{sábado}\\
\hline
\text{1ª semana}& \textcolor{blue}{*}\,\,\,^{\boxed{1}} &\textcolor{red}{*}\,\,\,^{\boxed{2}} & \textcolor{red}{*}\,\,\,^{\boxed{3}}& \textcolor{red}{*}\,\,\,^{\boxed{4}}&\textcolor{red}{*}\,\,\,^{\boxed{5}} &\textcolor{#58E817}{*}\,\,\,^{\boxed{6}} &\textcolor{red}{*}\,\,\,^{\boxed{7}}\\
\hline
\text{2ª semana}& \textcolor{blue}{*}\,\,\,^{\boxed{8}} &\textcolor{red}{*}\,\,\,^{\boxed{9}} & \textcolor{red}{*}^{\boxed{10}}& \textcolor{red}{*}^{\boxed{11}}&\textcolor{red}{*}^{\boxed{12}} &\textcolor{#58E817}{*}^{\boxed{13}} &\textcolor{red}{*}^{\boxed{14}}\\
\hline
\text{3ª semana}& \textcolor{blue}{*}^{\boxed{15}} &\textcolor{red}{*}^{\boxed{16}} & \textcolor{red}{*}^{\boxed{17}}& \textcolor{red}{*}^{\boxed{18}}&\textcolor{red}{*}^{\boxed{19}} &\textcolor{#58E817}{*}^{\boxed{20}} &\textcolor{red}{*}^{\boxed{21}}\\
\hline
\text{4ª semana}& \textcolor{blue}{*}^{\boxed{22}} &\textcolor{red}{*}^{\boxed{23}} & \textcolor{red}{*}^{\boxed{24}}& \textcolor{red}{*}^{\boxed{25}}&\textcolor{red}{*}^{\boxed{26}} &\textcolor{#58E817}{*}^{\boxed{27}} &\textcolor{red}{*}^{\boxed{28}}\\
\hline
\text{5ª semana}& \textcolor{blue}{*}^{\boxed{29}} & & & & & &\\
\hline\end{array}[/tex]

Observe que, mesmo que o mês tivesse [tex]31[/tex] dias, não teriam ocorrido cinco sextas-feiras nesse mês.

[tex]\begin{array}{r|r|r|r|r|r|r|r}
&\text{domingo} &\text{segunda} &\,\text{terça}\, &\text{quarta} &\text{quinta} &\text{ sexta }&\text{sábado}\\
\hline
\text{1ª semana}& \textcolor{blue}{*}\,\,\,^{\boxed{1}} &\textcolor{red}{*}\,\,\,^{\boxed{2}} & \textcolor{red}{*}\,\,\,^{\boxed{3}}& \textcolor{red}{*}\,\,\,^{\boxed{4}}&\textcolor{red}{*}\,\,\,^{\boxed{5}} &\textcolor{#58E817}{*}\,\,\,^{\boxed{6}} &\textcolor{red}{*}\,\,\,^{\boxed{7}}\\
\hline
\text{2ª semana}& \textcolor{blue}{*}\,\,\,^{\boxed{8}} &\textcolor{red}{*}\,\,\,^{\boxed{9}} & \textcolor{red}{*}^{\boxed{10}}& \textcolor{red}{*}^{\boxed{11}}&\textcolor{red}{*}^{\boxed{12}} &\textcolor{#58E817}{*}^{\boxed{13}} &\textcolor{red}{*}^{\boxed{14}}\\
\hline
\text{3ª semana}& \textcolor{blue}{*}^{\boxed{15}} &\textcolor{red}{*}^{\boxed{16}} & \textcolor{red}{*}^{\boxed{17}}& \textcolor{red}{*}^{\boxed{18}}&\textcolor{red}{*}^{\boxed{19}} &\textcolor{#58E817}{*}^{\boxed{20}} &\textcolor{red}{*}^{\boxed{21}}\\
\hline
\text{4ª semana}& \textcolor{blue}{*}^{\boxed{22}} &\textcolor{red}{*}^{\boxed{23}} & \textcolor{red}{*}^{\boxed{24}}& \textcolor{red}{*}^{\boxed{25}}&\textcolor{red}{*}^{\boxed{26}} &\textcolor{#58E817}{*}^{\boxed{27}} &\textcolor{red}{*}^{\boxed{28}}\\
\hline
\text{5ª semana}& \textcolor{blue}{*}^{\boxed{29}} & \textcolor{red}{?}^{\boxed{30}} & \textcolor{red}{?}^{\boxed{31}} & & & &\\
\hline\end{array}\\
\,\\[/tex]

Assim, o primeiro domingo do mês em questão não ocorreu na primeira semana do mês.

• Vamos supor agora que o primeiro domingo do mês em questão tivesse ocorrido na segunda semana do mês. Como ocorreram cinco domingos no mês, a distribuição mínima dos dias da semana nesse mês seria a mostrada na tabela abaixo.

[tex]\begin{array}{c|c|c|c|c|c|c|c}
&\text{domingo} &\text{segunda} & \text{ terça } &\text{quarta} &\text{quinta} &\text{ sexta }&\text{sábado}\\
\hline
\text{1ª semana}& & & & & & &\\
\hline
\text{2ª semana}& \textcolor{blue}{*} & \textcolor{red}{*} & \textcolor{red}{*}& \textcolor{red}{*}& \textcolor{red}{*} & \textcolor{#58E817}{*} & \textcolor{red}{*}\\
\hline
\text{3ª semana}&\textcolor{blue}{*} & \textcolor{red}{*} & \textcolor{red}{*}& \textcolor{red}{*}& \textcolor{red}{*} & \textcolor{#58E817}{*} & \textcolor{red}{*}\\
\hline
\text{4ª semana}& \textcolor{blue}{*} & \textcolor{red}{*} & \textcolor{red}{*}& \textcolor{red}{*}& \textcolor{red}{*} & \textcolor{#58E817}{*} & \textcolor{red}{*}\\
\hline
\text{5ª semana}& \textcolor{blue}{*} & \textcolor{red}{*} & \textcolor{red}{*}& \textcolor{red}{*}& \textcolor{red}{*} & \textcolor{#58E817}{*} & \textcolor{red}{*}\\
\hline
\text{6ª semana}& \textcolor{blue}{*} & & & & & &\\
\hline\end{array}[/tex]

Como ocorreram cinco sextas-feiras, o mês teria [tex]31[/tex] dias e a distribuição dos dias da semana nesse mês seria a mostrada na tabela abaixo.

[tex]\begin{array}{r|r|r|r|r|r|r|r}
&\text{domingo} &\text{segunda} & \text{ terça } &\text{quarta} &\text{quinta} &\text{ sexta }&\text{sábado}\\
\hline
\text{1ª semana}& & & & & & \textcolor{#58E817}{*}\,\,\,^{\boxed{1}} &\textcolor{red}{*}\,\,\,^{\boxed{2}} \\
\hline
\text{2ª semana}& \textcolor{blue}{*}\,\,\,^{\boxed{3}}& \textcolor{red}{*}\,\,\,^{\boxed{4}}&\textcolor{red}{*}\,\,\,^{\boxed{5}} &\textcolor{red}{*}\,\,\,^{\boxed{6}} &\textcolor{red}{*}\,\,\,^{\boxed{7}}&\textcolor{#58E817}{*}\,\,\,^{\boxed{8}} &\textcolor{red}{*}\,\,\,^{\boxed{9}} \\
\hline
\text{3ª semana}& \textcolor{blue}{*}^{\boxed{10}}& \textcolor{red}{*}^{\boxed{11}}&\textcolor{red}{*}^{\boxed{12}} &\textcolor{red}{*}^{\boxed{13}} &\textcolor{red}{*}^{\boxed{14}}& \textcolor{#58E817}{*}^{\boxed{15}} &\textcolor{red}{*}^{\boxed{16}}\\
\hline
\text{4ª semana} & \textcolor{blue}{*}^{\boxed{17}}& \textcolor{red}{*}^{\boxed{18}}&\textcolor{red}{*}^{\boxed{19}} &\textcolor{red}{*}^{\boxed{20}} &\textcolor{red}{*}^{\boxed{21}}&\textcolor{#58E817}{*}^{\boxed{22}}&\textcolor{red}{*}^{\boxed{23}}\\
\hline
\text{5ª semana}& \textcolor{blue}{*}^{\boxed{24}}& \textcolor{red}{*}^{\boxed{25}}&\textcolor{red}{*}^{\boxed{26}} &\textcolor{red}{*}^{\boxed{27}} &\textcolor{red}{*}^{\boxed{28}}& \textcolor{#58E817}{*}^{\boxed{29}} & \textcolor{red}{*}^{\boxed{30}}\\
\hline
\text{6ª semana} & \textcolor{blue}{*}^{\boxed{31}} & & & &\\
\hline\end{array}\\
\,\\[/tex]

Neste caso, o mês teria [tex]31[/tex] dias e teriam ocorrido quatro quintas-feiras nesse mês.

• Finalmente, observe que não podemos ter o primeiro domingo do mês ocorrendo a partir da segunda semana: de fato, devemos ter pelo menos um dia na primeira semana, o que implica em termos a segunda semana cheia e, consequentemente, um domingo estaria nessa semana.

[tex]\begin{array}{c|c|c|c|c|c|c|c}
&\text{domingo} &\text{segunda} & \text{ terça } &\text{quarta} &\text{quinta} &\text{ sexta }&\text{sábado}\\
\hline
\text{1ª semana}& \textcolor{red}{?}& \textcolor{red}{?}& \textcolor{red}{?}& \textcolor{red}{?}& \textcolor{red}{?}& \textcolor{red}{?}& \textcolor{red}{*}\\
\hline
\text{2ª semana}&\textcolor{red}{*} &\textcolor{red}{*} &\textcolor{red}{*} &\textcolor{red}{*} &\textcolor{red}{*} &\textcolor{red}{*} &\textcolor{red}{*}\\
\hline
\text{3ª semana}& \textcolor{blue}{*} & \textcolor{red}{*} & \textcolor{red}{*}& \textcolor{red}{*}& \textcolor{red}{*} & \textcolor{#58E817}{*} & \textcolor{red}{*}\\
\hline
\end{array}[/tex]
[tex]\dots[/tex]

Dessa forma, temos uma única situação que atende às hipóteses do problema. E nesse caso, conforme exposto, temos um mês com [tex]31[/tex] dias e com [tex]\fcolorbox{black}{#eee0e5}{$ \text{quatro quintas-feiras}$}.[/tex]

Solução elaborada pelos Moderadores do Blog.