The squares of an infinite table are numbered as follows: in the 0th row and 0th column we put 0, and then in every other square we put the smallest non-negative integer that does... The squares of an infinite table are numbered as follows: in the 0th row and 0th 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 1607th row and 1989th column? Can you generalize?
Understand the Problem
This question involves understanding a pattern in how numbers are placed in an infinite table based on specific rules. The task is to find the number that will appear in the 1607th row and 1989th column, and to generalize the approach for finding the numbers in any given row and column.
Answer
$4$. The number in the $r$-th row and $c$-th column is $(r + c) \pmod{8}$.
Answer for screen readers
The number that will appear in the 1607th row and 1989th column is 4. The general formula for the value in the $r$-th row and $c$-th column is $(r + c) \pmod{8}$.
Steps to Solve
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Observe the pattern in the table
We observe the given values to discern a pattern in the placement of numbers of the infinite table. Note that the value in the $i$-th row and $j$-th column appears to be $i+j$ modulo 8. This pattern is fundamental to solving the problem.
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Calculate the value at the specified location
We are asked to find the number in the 1607th row and 1989th column. According to the pattern, we calculate the sum of the row and column indices modulo 8: $$ (1607 + 1989) \pmod{8} $$
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Compute the sum
$$ 1607 + 1989 = 3596 $$
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Calculate the modulo
To find the remainder when 3596 is divided by 8, we perform the division: $$ 3596 \div 8 = 449 \text{ with a remainder of } 4 $$ Therefore, $3596 \equiv 4 \pmod{8}$.
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Generalize the solution.
For any row $r$ and column $c$, the number in that location is $(r + c) \pmod{8}$.
The number that will appear in the 1607th row and 1989th column is 4. The general formula for the value in the $r$-th row and $c$-th column is $(r + c) \pmod{8}$.
More Information
The pattern is based on the property of 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. This leads to the observed repeating pattern of length 8.
Tips
A common mistake is to miscalculate the modulo operation. Another mistake is to not recognize that the pattern repeats every 8 numbers, leading to an incorrect generalization.
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