1. On a grid paper, a rectangle 200 cells wide and 100 cells tall is drawn. It is filled cell by cell, starting from the top-left corner and following a spiral path (turning right when reaching an edge or a filled part, see Fig. 8). Which cell will be filled last? (Specify the number of its row and column. For example, the bottom-right cell is in the 100th row and 200th column.)

![](https://cdn.mathpix.com/cropped/2024_05_21_6455f48598646f41a890g-048.jpg?height=164&width=294&top_left_y=1307&top_left_x=951)

Fig. 8
1. First, the outer layer of cells will be shaded, after which a rectangle of $98 \times 198$ cells will remain. This rectangle will also be shaded in a spiral; after shading its outer layer, a rectangle of $96 \times 196$ cells will remain, and so on.

Let's see what will happen after shading 49 layers (Fig. 130). The top 49 rows and the bottom 49 rows will be shaded, meaning that the unshaded cells will only be in the 50th and 51st rows. Similarly, the left 49 columns (with numbers $1-49$) and the right 49 columns (with numbers $152-200$) will be shaded. Thus, the unshaded cells will be those located at the intersection of columns numbered $50-151$ and rows numbered 50-51 (this is a rectangle of $2 \times 102$). The last cell to be shaded will be the bottom-left cell of this rectangle, located in the $51$-st row and 50th column.

![](https://cdn.mathpix.com/cropped/2024_05_21_6455f48598646f41a890g-279.jpg?height=327&width=504&top_left_y=305&top_left_x=741)

Fig. 130