B4. On each of the 10,000 fields of a $100 \times 100$ chessboard, there is a number. On the top row, from left to right, the numbers are 0 to 99. In the left column, from top to bottom, the numbers are 0 to 99. The sum of four numbers in a $2 \times 2$ block is always 20. What number is in the bottom right corner of the board?
B4. -178

Color 10,000 squares according to a chessboard pattern: the squares at the top left and bottom right are white, and the squares at the bottom left and top right are black. Consider the $99 \cdot 99=9801$ squares of $2 \times 2$. Call the 4901 squares where the top left square is white the white squares, and the 4900 squares where the top left square is black the black squares. Add up the four numbers within each white square and then add up those 4901 results (some squares are counted twice!). Call the result $W$. Do the same for the black squares, with the result $Z$. Since the number of white squares is one more than the number of black squares, we have $W-Z=20$.

Now consider how often each square is included in total. The squares in the four corners each count in only one white square. The other squares on the edges each count in exactly one white and one black square. The remaining squares each count in exactly two black and two white squares. In the total $W-Z$, only the four corner squares count, each exactly once. For the number $x$ in the bottom right square, we find that $x+0+99+99=W-Z=20$, so $x=-178$.