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checkerboard satisfy the following conditions:
checkers have been placed on the board.
, where the checkers are the vertices and the edges are every pair of checkers sharing a side.
as the circuit rank of 
as the number of checkers placed on the board.
(this is the most naive upper bound).
![$5\times x - \text{[sum of degrees in G]} \ge n^2.$](http://latex.artofproblemsolving.com/9/e/3/9e3028fa438f28ec1b1c921e0d1a62d0fae44ad9.png)
(see lemma below).
where 
and is equal to 0 if 
The sum of degrees of a connected graph 
is 
where 
is the sum of ranks of the spanning tree created by decircuiting the graph 
is the sum of ranks removed from 
checkers on the board, sitting on squares 
. We can assume WLOG that 
, then 
shares a side with at least one of 
. (If such an ordering were impossible, then there would be an 
by a sequence of squares with checkers on them, violating condition (b).)
in that order, counting the number of good squares after each
step. When we put a checker on 
, we increase the number of good squares by at most 
, since 
of its at most 
neighbors already has a checker on it.
. In order to satisfy condition (a), we have to make each of the 
squares good. In other words, we need 
, or ![\[k \geq \frac{n^2 - 2}{3}.\]](http://latex.artofproblemsolving.com/f/6/5/f65aca44a85215dab73e6359dcc561c0adcc99aa.png)
