Problem: Let $n$ be a positive integer.
Find the greatest possible integer $m$, in terms of $n$,
with the following property: a table with $m$ rows and $n$ columns can be
filled with real numbers in such a manner that for any
two different rows $\left[ a_1, a_2, \dots, a_n \right]$
and $\left[ b_1, b_2, \dots, b_n \right]$ the following holds:
\[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,
  \dots,\left| {{a_n} - {b_n}} \right|} \right) = 1. \]

Solution: \paragraph{Answer.} The largest $m$ is $m(n) = 2^n$.

\paragraph{Construction.} Choose the table with all $2^n$ binary vectors.

\paragraph{Bound.}
We will proceed by induction on $n$, with the base case $n=1$ being immediate.
Take the first column, and assume the entries lie in some interval $I = [r, r+1]$.
Then the rows with first entry equal to $r$
form an $(n-1)$-table, hence at most $m(n-1)$ of them.
The rows with first entry not equal to $r$
also form an $(n-1)$-table, hence at most $m(n-1)$ of them.
So
\[ m(n) \le \underbrace{m(n-1)}_{\text{first entry $r$}}
  + \underbrace{m(n-1)}_{\text{first entry not $r$}} = 2m(n-1) \]
and we're done by induction.