It's a bit tricky to see Pigeon Hole Principle. Here we note that the total game play is not to exceed $11\times 12=132$ games. Now, if we denote $s_i$ to be the total games played after the $i$-th day. Then look at the sequence of $\color{red}{154}$ numbers:
$$s_1,s_2,\dots, s_{77}, s_{1}+21,\dots, s_{76}+21,s_{77}+21$$
Now these are the "pigeons" whereas their values are "holes". Clearly, the maximal value (i.e. the number of holes) is $132+21=\color{red}{153}$.
The primes below 10 are 4:
$2,3,5,7$
So each of your numbers has the form:
$2^a \cdot 3^b \cdot 5^c \cdot 7^d$
There 16 possible tuples (a,b,c,d) when each of $a,b,c,d$ is viewed modulo 2.
We define each pigeonhole as the sequence $(A,B,C,D)$ where A,B,C,D are the residues of a,b,c,d respectively when divided by 2. There are 16 possible pigeonholes ($2 \cdot 2 \cdot 2 \cdot 2$).
This means you can find two numbers $X,Y$ among those 17 such that their tuples
$a_1,b_1,c_1,d_1$ and $a_2,b_2,c_2,d_2$
are such that
$a_1$ and $a_2$ are both odd or both even
$b_1$ and $b_2$ are both odd or both even
$c_1$ and $c_2$ are both odd or both even
$d_1$ and $d_2$ are both odd or both even
Now multiply X and Y and you get a square because
$a_1 + a_2$, $b_1 + b_2$, $c_1 + c_2$, $d_1 + d_2$
will all be even.
Best Answer
We show that
We prove by induction on $k$. For $k = 0$ there is nothing to prove.
Now suppose we have $n + k$ rows and at most $n + 2k$ zeros. Without loss of generality, we may assume that there are exactly $n + 2k$ zeros (otherwise, we pretend that some of the ones were zeros, and proceed as follows).
Since there are $n + 2k$ zeros and only $n + k$ rows, pigeon hole principle tells us that there exists one row that contains at least $2$ zeros. We remove that row.
Now there remains $n + (k - 1)$ rows and at most $n + 2(k - 1)$ zeros, so the induction hypothesis finishes the rest.
For $k = n$, we have shown that if there are $3n$ zeros in $2n$ rows, then we may remove $n$ rows such that there remains at most $n$ zeros.
Then simply remove all the columns containing at least a zero.