Question

A square with an integer side length is cut into $10$ squares, all of which have integer side length and at least $8$ of which have area $1.$ What is the smallest possible value of the length of the side of the original square?

$3$

$4$

$5$

$6$

$7$

Answer: B

Concepts:tiling extremal argument

Difficulty rating: 1540

Small Hint:

A side length of $3$ gives area too small for ten integer squares.

Big Hint:

A $4$ by $4$ square can be cut as shown.

Video solution:

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Written solution:

Since all $10$ squares have positive integer side lengths, each has area at least $1$ . Their total area is therefore at least $10$ . A square with integer side length at most $3$ has area at most $3^2=9$ , so its side length cannot be less than $4$ .

The following configuration cuts a $4\times4$ square into ten integer-sided squares, eight of which are unit squares. Therefore side length $4$ is attainable and is the minimum.

Thus, the answer is B .

2012 AMC 8 Problem 17