MathDB

We number the columns of an

n\times n

-board from

1

n

. In each cell, we place a number. This is done in such a way that each row precisely contains the numbers

1

n

(in some order), and also each column contains the numbers

1

n

(in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case

n = 3

.
[asy] unitsize(0.6 cm);
int i;
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray(0.8)); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8));
for (i = 0; i <= 3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); }
label("

1

", (0.5,3.5)); label("

2

", (1.5,3.5)); label("

3

", (2.5,3.5)); label("

3

", (0.5,2.5)); label("

1

", (1.5,2.5)); label("

2

", (2.5,2.5)); label("

1

", (0.5,1.5)); label("

2

", (1.5,1.5)); label("

3

", (2.5,1.5)); label("

2

", (0.5,0.5)); label("

3

", (1.5,0.5)); label("

1

", (2.5,0.5)); [/asy]
(a) Suppose that

n = 5

. Can the numbers be placed in such a way that each row contains the same number of grey cells? (b) Suppose that

n = 10

. Can the numbers be placed in such a way that each row contains the same number of grey cells?

Problem Statement