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 understand the pattern of numbers.
Let's start with the first few rows and columns to see if we can find a pattern.
Row 1: 0, 1, 2, 3, 4, ...
Row 2: 1, 3, 5, 7, 9, ...
Row 3: 2, 6, 10, 14, 18, ...
From this, we can observe that the number in the first row is simply the index number (0, 1, 2, 3, ...) itself. In the second row, we can see that the numbers are obtained by adding 2 to the previous number. Similarly, in the third row, we add 4 to the previous number, and so on.
Based on this pattern, we can determine the number in the 1000th row and 700th column by following these steps:
1. Find the number in the first row of the same column. Since the first row has the same number as the index, the number in the 1000th row and 700th column will be 700.
In general, to find the number in the nth row and mth column, we can use the formula:
Number = (n-1) + (m-1) * 2
In this case, substituting n = 1000 and m = 700 into the formula, we get:
Number = (1000-1) + (700-1) * 2
= 999 + 1398
= 2397
Therefore, the number that will appear in the 1000th row and 700th column is 2397.
This formula can be used to find the number in any row and column of the infinite chessboard based on the given pattern.