MathDB

Given positive integers

n

and

k \leq n

. Consider an equilateral triangular board with side

n

, which consists of circles: in the first (top) row there is one circle, in the second row there are two circles,

\ldots

, in the bottom row there are

n

circles (see the figure below). Let us place checkers on this board so that any line parallel to a side of the triangle (there are

3n

such lines) contains no more than

k

checkers. Denote by

T(k, n)

the largest possible number of checkers in such a placement. https://i.ibb.co/bJjjK1M/Image2.jpg a) Prove that the following upper bound is true:

T(k,n) \leq \lfloor \frac{k(2n+1)}{3} \rfloor

b) Find

T(1,n)

and

T(2,n)

D. Zmiaikou

Problem Statement