6. The squares of an infinite chessboard are numbered as follows: in the first row and first column
we put 0, and then in every other square we put the smallest non-negative integer that does not
appear anywhere below it in the same column or anywhere to the left of it in the same row.
What number will appear in the 1000th row and 700th column? Can you generalize?
To find the number that will appear in the 1000th row and 700th column of the infinite chessboard, we need to follow the given pattern:
1. Start with the first cell in the first row and first column, which is labeled as 0.
2. In the first row, fill in the numbers from left to right such that each number is the smallest non-negative integer that has not appeared in any previous column in the same row.
3. In the remaining rows, follow the same pattern but make sure the numbers have not appeared in any previous column or row.
4. Continue this process until we reach the desired row and column.
Let's calculate the number that will appear in the 1000th row and 700th column:
- In the first row, the numbers will be filled from left to right as 0, 1, 2, 3, ...
- In the second row, the first column will be filled with the smallest non-negative integer that has not appeared in the previous row (row 1) or any previous column (left of the current column in the same row). So, the number in the second row, first column will be 1001.
- Continuing this pattern, we can calculate the number that will appear in the 1000th row, 700th column.
Generalizing the process, we can use the following equation to calculate the number in the nth row and mth column:
Number in nth row, mth column = (n-1) * (m+1) + (m-1)
Therefore, the number that will appear in the 1000th row and 700th column is:
Number in 1000th row, 700th column = (1000-1) * (700+1) + (700-1)
= 999 * 701 + 699
= 699699+ 699
= 700398
So, the number that will appear in the 1000th row and 700th column of the infinite chessboard is 700398.