To glaze 15 windows of various sizes and shapes, 15 pieces of glass have been prepared exactly to fit the windows (the windows are such that each window should have one piece of glass). The glazier, not knowing that the glasses are matched, works as follows: he approaches the next window and tries the unused glasses until he finds one that is large enough (i.e., either exactly fitting or one from which a fitting piece can be cut), if no such glass is found, he moves to the next window, and so on, until he has gone through all the windows. It is not allowed to assemble a piece of glass from several parts. What is the maximum number of windows that can remain unglazed?
First, let's show that if at any moment there are no fewer than 8 windows (and, accordingly, no fewer than 8 panes), then a pane for one of the remaining windows can be found. Indeed, no more than seven panes have been used, so at least one of the eight panes, intended for the eight remaining windows, remains. This one can be inserted into its "own" window. Therefore, more than seven windows cannot remain unglazed.

Now, let's show that seven windows can remain unglazed. This will happen, for example, in the following case. The panes have such sizes that any one of them can be obtained from another by cutting (note that the problem statement does not require this). Thus, all panes can be ordered from the smallest to the largest so that any smaller one can be "cut" from a larger one.

The windows can be similarly ordered. Suppose the glazier approaches the windows in the order of increasing size (from the smallest to the largest), while selecting panes in the order of decreasing size. In this case, he will manage to glaze the 8 smallest windows, the eighth window will get its "own" pane. The 7 largest windows will remain unglazed, and the 7 smallest panes will remain unused.