MathDB

A calendar is a (finite) rectangular grid. A calendar is valid if it satisfies the following conditions:
(i) Each square of the calendar is colored white or red, and there are exactly 10 red squares.
(ii) Suppose that there are

N

columns of squares in the calendar. Then if we fill in the numbers

1,2,\ldots

from the top row to the bottom row, and within each row from left to right, there do not exist

N

consecutive numbers such that the squares they are in are all white.
(iii) Suppose that there are

M

rows of squares in the calendar. Then if we fill in the numbers

1,2,\ldots

from the left-most column to the right-most column, and within each column from bottom to top, there do not exist

M

consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by

90^{\circ}

, the resulting calendar still satisfies (ii).
How many different kinds of valid calendars are there?
(Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the

10

red squares are at different locations.)

Problem Statement