MathDB

One hundred empty glasses are arranged in a

10 \times 10

array. Now we pick

a

of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick

b

of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as it is.
[*] Determine all possible combinations of values for

a

and

b

so that exactly half of the glasses contain green liquid at the end. [*] Is it possible that precisely one quarter of the glasses contain green liquid at the end?

Problem Statement