7. A monkey has filled in a 3 x 3 grid with the numbers 1, 2, ..., 9. A cat writes down the three numbers obtained by multiplying the numbers in each horizontal row. A dog writes down the three numbers obtained by multiplying the numbers in each vertical column. Can the monkey fill in the grid in such a way that the cat and dog obtain the same lists of three numbers? What if the monkey writes the numbers 1, 2, ..., 25 in a 5 x 5 grid? Or 1,2, ..., 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that it is possible or any conditions that guarantee that it is impossible for the monkey to write the numbers 1,2,..., n² in an n x n grid so that the cat and the dog obtain the same lists of numbers?
7. A monkey has filled in a 3 x 3 grid with the numbers 1, 2, ..., 9. A cat writes down the three numbers obtained by multiplying the numbers in each horizontal row. A dog writes down the three numbers obtained by multiplying the numbers in each vertical column. Can the monkey fill in the grid in such a way that the cat and dog obtain the same lists of three numbers? What if the monkey writes the numbers 1, 2, ..., 25 in a 5 x 5 grid? Or 1,2, ..., 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that it is possible or any conditions that guarantee that it is impossible for the monkey to write the numbers 1,2,..., n² in an n x n grid so that the cat and the dog obtain the same lists of numbers?
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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Transcribed Image Text:7. A monkey has filled in a 3 x 3 grid with the numbers 1, 2,...,9. A cat writes down the
three numbers obtained by multiplying the numbers in each horizontal row. A dog writes
down the three numbers obtained by multiplying the numbers in each vertical column.
Can the monkey fill in the grid in such a way that the cat and dog obtain the same lists
of three numbers? What if the monkey writes the numbers 1, 2, . . . , 25 in a 5 × 5 grid? Or
1, 2, ..., 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that it
is possible or any conditions that guarantee that it is impossible for the monkey to write
the numbers 1, 2, ..., n² in an n x n grid so that the cat and the dog obtain the same
lists of numbers?
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