275. What is the minimum number of chips that need to be placed on the fields of a chessboard of size

a) $8 \times 8$ cells,

b) $n \times n$ cells

so that on each line passing through the center of any field and parallel to any side or diagonal of the board, there is at least one chip? (Chips are placed in the centers of the fields.)
275. Answer: a) for $n=8-16$ chips; b) for $n$ even $\rightarrow$ $2 n$, for $n$ odd $-2 n+1$.

The arrangement of such a number of chips is clear from Fig. $113, a, 6$. The proof that a smaller number cannot suffice is simpler for even $n$; on each line parallel to one diagonal, there should be a chip, and on the diagonal itself - two (in the corners).

Another proof: on each dashed line shown in the figures, there should be a chip. This proof is adapted for the case of odd $n$ (Fig. $113, b$), besides $2 n-2$ dashed lines (each with a chip); it is necessary to consider six more lines connecting the centers of cells $A, B, C, D$; at least 3 more chips are needed for these.