Red Perimeter 1 – Extend the Pattern

Probability Level 1

Given n × n n \times n squares made out of 1 × 1 1 \times 1 unit squares with the perimeter of the n × n n \times n square painted red, some of the 1 × 1 1 \times 1 squares will be painted on 2 sides, some on only 1 side, and some won't be painted at all.

How many unit squares will be painted red on only 1 side in a 4 × 4 4 \times 4 square (not pictured)?

This problem is part of Arron's set The Red Perimeter .


The answer is 8.

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Arron Kau by Arron Kau

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5 solutions

Discussions for this problem are now closed

Hayley Brooks
May 14, 2014

A 4x4 square has a perimeter of 16. On each side the two corners have red on two of their sides, meaning that half of each side of the big square, have a square that is only red on one side. 16/2=8

15 Helpful 0 Interesting 0 Brilliant 0 Confused

total sqrs touchingred line of perimeter is 12. four corner sqrs are red on two bsides .remaining 8 are red lined on one side.

8 Helpful 0 Interesting 0 Brilliant 0 Confused
Kasper Knudsen
May 20, 2014

Given n x n squares the perimeter is 4n. Since the corner squares should not be counted number twice the number of unit squares touching the perimeter is 4n-4. Only the corner squares are not painted on 1 side, so they must be subtracted. The number of unit squares that will be painted red on only 1 side is given 4n-8=4(n-2), and the solution is 4(4-2)=4*2=8.

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Mark Sinsheimer
May 15, 2014

N squares per side less 2 for each corner with two sides. 4 sides to a square. For a NxN square, number of one-sided-red-painted-square X is 4(N-2). If N=4 => X = 8

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Atul Vaibhav
Jun 25, 2014

A 4X4 square will have 2 squares on each side which will be painted on only 1 side .Therefore total no. of corresponding squares will be 8

2 Helpful 0 Interesting 0 Brilliant 0 Confused

but a cube has 6 sides

Anuj Shikarkhane - 6 years, 11 months ago

bt here it is plane square not cube

Atul Vaibhav - 6 years, 10 months ago

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