diff --git a/Advanced/Pidgeonhole Problems/main.tex b/Advanced/Pidgeonhole Problems/main.tex index c62de90..8d3f40b 100755 --- a/Advanced/Pidgeonhole Problems/main.tex +++ b/Advanced/Pidgeonhole Problems/main.tex @@ -35,7 +35,7 @@ \problem{} - 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 $n$. \begin{solution} @@ -55,7 +55,7 @@ \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} 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 $b$. \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 \begin{itemize}