Problem 6.6. Ksyusha placed a chip in one of the cells of a $4 \times 4$ square. She moved the chip from cell to cell within this square, each time moving to an adjacent cell by side, and visited each cell exactly once. In each cell, she recorded the move number on which the chip landed in that cell (the left image). She did this again, but recorded only 3 moves (the right image). Find out which numbers should be in the marked cells.

| 16 | 15 | 14 | 13 |
| :---: | :---: | :---: | :---: |
| 3 | 2 | 11 | 12 |
| 4 | 1 | 10 | 9 |
| 5 | 6 | 7 | 8 |


| A |  | 2 |  |
| :---: | :---: | :---: | :---: |
|  | B | 5 |  |
|  | 13 | C |  |
|  |  |  | D |

(a) Instead of the letter $A$
(1) the number 6 stood.
(b) Instead of the letter $B$
(2) the number 8 stood.
(c) Instead of the letter $C$
(3) the number 14 stood.
(d) Instead of the letter $D$
(4) the number 16 stood.
Answer: $\mathrm{a} 4, \mathrm{~b} 3, \mathrm{c} 1, \mathrm{~d} 2$

Solution. We can start either to the left or to the right of the cell with number 2. If we start from the cell to the right, the sequence of cells up to number 5 is uniquely determined. By enumeration, we can verify that only one route remains, and it does not land on the cell with number 13 at the corresponding move. Therefore, we cannot start from the right.

| $\mathbf{A}$ |  | $\mathbf{2}$ |  |
| :---: | :---: | :---: | :---: |
|  | $\mathbf{B}$ | 5 |  |
|  | $\mathbf{1 3}$ | $\mathbf{C}$ |  |
|  |  |  | $\mathbf{D}$ |
|  |  |  |  |

Thus, we start from the cell to the left of 2. Reaching cell 5, we can turn into cells $B$ or $C$. If we turn into cell $B$, the next move must be into cell 13 (otherwise, the entire square cannot be covered), which contradicts the condition. Therefore, cell $C$ has number 6; the route is then uniquely constructed.

| $\mathbf{A}$ | - | 2 |
| :---: | :---: | :---: |
| 4 | B | 5 |
| $\square$ | 13 | C |
| L |  |  |