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17*17 table with some cells colored in black, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 2, problem 6.

January 17, 2009

floor functioninequalitiesceiling functioncombinatorics proposedcombinatorics

Problem Statement

In a checked

17\times 17

table,

n

squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least

6

of the squares in some line are black, then one can paint all the squares of this line in black. Find the minimal value of

n

such that for some initial arrangement of

n

black squares one can paint all squares of the table in black in some steps.