## Nine Dates

You are looking at a printed calendar for a given month. The weeks are in rows. Each day is a square on the calendar. You draw a three-by-three square around nine dates. You notice that the sum of these $9$ dates is divisible by $13$. What is the date in the lower-right corner of your three-by-three square?

Source: mathcontest.olemiss.edu 8/27/2012

SOLUTION
There are eleven three-by-three squares that contain $9$ dates.
$1\quad\;\;\;2\quad\;\;\;3$
$8\quad\;\;\;9\quad\;10$
$15\quad 16\quad\;17$
Sum $=81$. NO, not divisible by $13$.

$2\quad\;\;\;3\quad\;\;\;4$
$9\quad\;10\quad\;\;11$
$16\quad 17\quad\;18$
Sum $=90$. NO.

$5\quad\;\;\;6\quad\;\;\;7$
$12\quad 13\quad\;14$
$19\quad 20\quad\;21$
Sum $=117$. YES, divisible by $13$. The lower-right corner number is $21$.

None of the eight remaining sums $126,135,144,153,180,189,198,207$ is divisible by $13$.

Answer: $21$.