Combinatorics · Double counting · Parity · Constructions
An \(n \times n\) table is to be filled with zeros and ones so that for every index \(i \in \{1, 2, \dots, n\}\) the number of ones in the \(i\)-th row and the number of ones in the \(i\)-th column differ by exactly \(1\) in absolute value.
Decide whether such a filling exists when
a) \(n = 2015\);
b) \(n = 2016\).
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Serbian Regional Competition (Okruzno takmicenje) 2016, high school grade I, category A, problem 4. Organized by the Mathematical Society of Serbia (DMS). Source