Problem number 17. Mark wants to fill in the cells on the diagram shown so that each cell contains either a cross or a circle. He also wants to ensure that there is no line of four consecutive identical symbols in any column, row, or diagonal. When he has completed the diagram, what will the column colored grey contain? To solve this problem, let's look at each of the rows, columns, and diagonals and see if there are any cases where we can uniquely determine a shape. For example, if we look at this highlighted diagonal in red, we can notice that it already has three Xs. To avoid having four Xs in a consecutive pattern on the diagonal, we have to fill an O in the blue spot marked on the diagram. Now, we can do the same logic with this column marked with the red line. We can see that in both cases we have to put an X, because if we put an O there, then we would have four Os in either the top four cells or the bottom four cells, and that cannot happen, so we have to put Xs in those locations. Now if we look at this diagonal, we see that there are three Xs already on the diagonal, so the fourth cell has to be an O. Similarly, we can see that in this row there are three Os already, so we have to put an X on either end, because if we put a circle there, then we will have four Os in a row. Now if we look at this column, we already have three Xs, so we have to put an O to fill out the column, and finally in this diagonal, we see that we already have three Os, so we have to put an X to avoid having four Os. In this row, we have the same issue where we have three Xs in a row, so now we have to put an O on either end to stop there from being four Xs. And now here, there are two lines to look at. If we look at the diagonal first, we already see that we have three Os, so we have to put an X there at the end of the diagonal. In the column, we see that we have three Os with a gap in the middle, and that gap has to be filled with an X to avoid having four Os. Now we're almost done. We can see that that diagonal here has three Xs, so we have to put an O at either end, and the row has three Xs with a gap in the middle, so that gap has to be filled with an O. And now the final step. We already know that this diagonal has three Os, so we have to fill an X in the final square. And now we just have to look at the gray column, and we see that it has two circles and four crosses, so that means the answer is B, two circles and four crosses.

grid filling

consecutive symbols

strategic placement

pattern prevention

crosses and circles