TITLE: 3 congruent shapes in a $5 \times 5$ square board
QUESTION [5 upvotes]: Given a $5 \times 5$ board of $25$ cells. Is it possible to color $24$ of them in one of three colors in such a way, that the resulting three shapes of the particular colors are congruent to each other?

REPLY [2 votes]: (1)
(2)
(3)
One of them have two corners. Then in the case of $(1)$, another shape must be two corners, too.
However $(2,1)(4,1)(2,5)(4,5)$ can't devide to three areas.
In the case of $(2)$, we can't draw $(3,3), (5,1)$ or $(5,5)$ into a color, satisfying the condition.
In the case of $(3)$, if we draw the color of $(3,3)$, we can't make congruent shapes.
Therefore three congruent shapes are impossible.