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Mark 2024-10-04 13:15:44 -07:00
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Advanced/Pidgeonhole Problems/main.tex
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@ -35,7 +35,7 @@
\problem{}<divisibledifference> \problem{}<divisibledifference>
You are given $n + 1$ integers. \par You are given $n + 1$ distinct integers. \par
Prove that at least two of them have a difference divisible by $n$. Prove that at least two of them have a difference divisible by $n$.
\begin{solution} \begin{solution}
@ -55,7 +55,7 @@
\problem{} \problem{}
You have an $8 \times 8$ chess board with two opposing corner squares cut off. You also have a set of dominoes, each of which is the size of two squares. Is it possible to completely cover the the board with dominos, so that none overlap nor stick out? You have an $8 \times 8$ chess board with two opposing corner squares cut off. You also have a set of dominoes, each of which is the size of two squares. Is it possible to completely cover the board with dominos, so that none overlap nor stick out?
\begin{solution} \begin{solution}
A domino covers two adjacent squares. Adjacent squares have different colors. \par A domino covers two adjacent squares. Adjacent squares have different colors. \par
@ -253,7 +253,7 @@
Choose $n + 1$ integers between $1$ and $2n$. Show that you must select two numbers $a$ and $b$ such that $a$ divides $b$. Choose $n + 1$ integers between $1$ and $2n$. Show that you must select two numbers $a$ and $b$ such that $a$ divides $b$.
\begin{solution} \begin{solution}
Split the the set $\{1, ..., 2n\}$ into classes defined by each integer's greatest odd divisor. There will be $n$ classes since there are $\frac{k}{2}$ odd numbers between $1$ and $n$. Because we pick $n + 1$ numbers, at least two will come from the same class---they will be divisible. \par Split the set $\{1, ..., 2n\}$ into classes defined by each integer's greatest odd divisor. There will be $n$ classes since there are $\frac{k}{2}$ odd numbers between $1$ and $n$. Because we pick $n + 1$ numbers, at least two will come from the same class---they will be divisible. \par
For example, if $n = 5$, our classes are For example, if $n = 5$, our classes are
\begin{itemize} \begin{itemize}