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Challenge 161: Counter Cover-Up

Can you find the smallest amount of counters you would need to cover this square?

I have some circular counters with radius 1cm.

What is the smallest number of counters I would need to use to completely cover a square with side 2cm? Or 3cm? Or n cm?

What about a rectangle measuring n by m cm?

Remember to justify your answers.

(The counters must remain whole – I am not allowed to cut them into smaller pieces!)