Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.

(Day 1, 3rd problem
author: Michal Rolínek)
1. **Identify the total number of edges on the chessboard:**
   - The $8 \times 8$ chessboard has $8$ rows and $8$ columns, creating $9$ horizontal and $9$ vertical lines of edges.
   - Each row has $8$ horizontal edges, and there are $9$ rows, so there are $8 \times 9 = 72$ horizontal edges.
   - Each column has $8$ vertical edges, and there are $9$ columns, so there are $8 \times 9 = 72$ vertical edges.
   - Therefore, the total number of edges is $72 + 72 = 144$.

2. **Identify the internal edges:**
   - Internal edges are those that are not on the boundary of the chessboard.
   - There are $7$ internal horizontal edges per row, and $8$ rows, so there are $7 \times 8 = 56$ internal horizontal edges.
   - Similarly, there are $7$ internal vertical edges per column, and $8$ columns, so there are $7 \times 8 = 56$ internal vertical edges.
   - Therefore, the total number of internal edges is $56 + 56 = 112$.

3. **Determine the number of dominoes in a partition:**
   - A $1 \times 2$ domino covers exactly $2$ squares.
   - The $8 \times 8$ chessboard has $64$ squares.
   - Therefore, the number of dominoes required to cover the entire chessboard is $\frac{64}{2} = 32$.

4. **Calculate the contribution of each partition to the sum of the numbers on the internal edges:**
   - Each domino covers exactly $2$ squares and thus uses exactly $1$ internal edge.
   - Since there are $32$ dominoes in each partition, each partition uses $32$ internal edges.
   - However, each internal edge is shared by two dominoes, so the total contribution of each partition to the sum of the numbers on the internal edges is $32 \times 2 = 64$.

5. **Calculate the total contribution of all partitions:**
   - The number of ways to partition the chessboard into $1 \times 2$ dominoes is given by the number of perfect matchings in the corresponding bipartite graph.
   - For an $8 \times 8$ chessboard, this number is known to be $12988816$.
   - Therefore, the total contribution to the sum of the numbers on the internal edges is $12988816 \times 64$.

6. **Find the last digit of the sum:**
   - We need to find the last digit of $12988816 \times 64$.
   - Calculate the last digit of $12988816$: it is $6$.
   - Calculate the last digit of $64$: it is $4$.
   - The last digit of the product of two numbers is the product of their last digits.
   - Therefore, the last digit of $12988816 \times 64$ is the last digit of $6 \times 4 = 24$, which is $4$.

The final answer is $\boxed{4}$.