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Problem 23 Easy

A rectangular grid of souares has rows and columns. Each souare has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from through into the squares. Horace fills the grid horizontally: he puts through in order from left to right into row , puts through into row in order from left to right, and continues similarly through row . Vera fills the grid vertically: she puts through in order from top to bottom into column , then through into column in order from top to bottom, and continues similarly through column . How many squares get two copies of the same number?

Answer:C

Solution

Let and be the numbers filled by Horace and Vera respectively at row , column .

Then:

For :

Let and :

where and .

This gives and for .

There are 11 solutions.