Figures of Constant Width on a Chessboard

Every row and column of the board has only two occupied squares. But if we also consider the diagonals with slope +1 or −1, we see that each of them contains either zero or two squares of the figure. We say that Figure 1 has constant width three by rows and columns, and that Figure 2 has constant width two by rows, columns, and diagonals. The first figure is easy to generalize in the sense that the figure formed by all the squares of a w × w chessboard has constant width w by rows and columns. On the other hand, if we also consider the two diagonal directions as in Figure 2, the problem of finding figures of constant width higher than two becomes considerably harder. Is it possible, for example, to find a nonempty figure on some n × n chessboard such that every row, column, and diagonal intersects it in zero or three squares? This is a fun problem to think about. We encourage the reader to try it before he or she continues reading. As we will show here, the answer turns out to be yes, even for widths higher than three. The impatient reader might want to take a look at Figures 6, 11, 12, and 13. In order to state our main result in a precise form, we introduce some terminology. We designate as a figure any set of squares in an n × n chessboard. A figure F has constant width w if every row, column, or diagonal intersects it in 0 or w squares. To be more exact, F is of type (n, k, w) if it has constant width w in a chessboard of size n × n and has kw squares. Observe that k is also the number of nonempty rows (or columns) in the chessboard.