Problem of the Week
Problem B and Solution
Quilt Designs for Barns

Problem

Sometimes people decorate their barns with quilts. These barn quilts are often designed in a square grid with a variety of colours and patterns, and they usually have symmetry in such a way that if they’re rotated they will look the same. In the \(6\) by \(6\) grids below, you are going to explore and create some barn quilt designs. In each part, you can add straight lines as needed.

1. What fraction of the barn quilt design below is black, and what fraction is white? In the blank grid, design another barn quilt using two colours and the same fractions.

   

   Hide/Reveal Description of Quilt for (a)

   The \(6\) by \(6\) quilt pattern is formed by nine \(2\) by \(2\) squares arranged into \(3\) rows and \(3\) columns. The quilt looks the same when rotated by a quarter turn.

   • The \(2\) by \(2\) square in the centre is coloured entirely black.
   • The four \(2\) by \(2\) squares in the corners are coloured entirely white.
   • On the other four \(2\) by \(2\) squares, the two diagonals are drawn dividing the square into four identical triangles. One pair of opposite triangles is coloured white and the other pair is coloured black.

   ---

2. What fraction of the barn quilt design below is black? Grey? White? In the blank grid, design another barn quilt using three colours and the same fractions.

   

   Hide/Reveal Description of Quilt for (b)

   The \(6\) by \(6\) quilt pattern is formed by nine \(2\) by \(2\) squares arranged into \(3\) rows and \(3\) columns. The quilt looks the same when rotated by a quarter turn.

   • The \(2\) by \(2\) square in the centre is coloured entirely white.
   • The four \(2\) by \(2\) squares in the corners are coloured entirely black.
   • On the other four \(2\) by \(2\) squares, the two diagonals are drawn dividing the square into four identical triangles. One pair of opposite triangles is coloured white and the other pair has one black and one grey.

   ---

3. What fraction of the following barn quilt design below is black? Grey? Striped? White? In the blank grid, design another barn quilt using four colours and the same fractions.

   

   Hide/Reveal Description of Quilt for (c)

   The \(6\) by \(6\) quilt pattern is formed by nine \(2\) by \(2\) squares arranged into \(3\) rows and \(3\) columns. The quilt looks the same when rotated by a half turn.

   • The four \(2\) by \(2\) squares in the corners each have one of the four smaller squares coloured grey and the other three coloured white.
   • On the other five \(2\) by \(2\) squares, the two diagonals are drawn dividing the square into four identical triangles.
     • In the \(2\) by \(2\) square in the centre, one pair of opposite triangles is coloured grey and the other pair is striped.
     • In the other four \(2\) by \(2\) squares, one pair of opposite triangles is striped and the other pair has one black and one white.

   ---

Solution

Note that a \(6\) by \(6\) grid consists of \(36\) smaller squares. There are also smaller triangles in the designs which each make up half of a smaller square. (These smaller triangles are formed when the two diagonals of a \(2\) by \(2\) square in the grid are drawn.)

1. The given barn quilt design has \(16\) white smaller squares plus \(16\) white smaller triangles, equivalent to \(16\div 2 = 8\) smaller squares. Therefore, the amount that is white is equivalent to \(16 +8 = 24\) smaller squares, and the fraction that is white is \(\frac{24}{36} = \frac{2}{3}\). Thus, the fraction that is black must be \(1-\frac{2}{3}=\frac{1}{3}\). Two barn quilt designs using two colours and the same fractions are given below.

   

2. The given barn quilt design has \(16\) black smaller squares plus \(8\) black smaller triangles, equivalent to \(8\div 2 = 4\) smaller squares. Therefore, the amount that is black is equivalent to \(16 + 4 = 20\) smaller squares, and the fraction that is black is \(\frac{20}{36} = \frac{5}{9}\). There are \(8\) grey smaller triangles, equivalent to a total of \(8\div 2 = 4\) grey smaller squares. Therefore, the fraction that is grey is \(\frac{4}{36} = \frac{1}{9}\) and the fraction of white is \(1- \frac{5}{9} - \frac{1}{9} = \frac{3}{9} = \frac{1}{3}\). Two barn quilt designs using three colours and the same fractions are given below.

   

3. The given barn quilt design has \(12\) white smaller squares plus \(8\) white smaller triangles, equivalent to \(8\div 2 = 4\) smaller squares. Therefore, the amount that is white is equivalent to \(12 + 4 = 16\) white smaller squares, and the fraction that is white is \(\frac{16}{36}= \frac{4}{9}\). There are \(8\) black smaller triangles, equivalent to a total of \(8\div 2 = 4\) black smaller squares. Therefore, the fraction that is black is \(\frac{4}{36} = \frac{1}{9}\). There are \(4\) smaller grey squares plus \(4\) grey smaller triangles, equivalent to \(4\div 2 = 2\) smaller squares. Therefore, the amount that is grey is equivalent to \(4 + 2 = 6\) grey smaller squares, and the fraction that is grey is \(\frac{6}{36}=\frac{1}{6}\). There are \(20\) striped smaller triangles, equivalent to \(20\div 2 =10\) striped smaller squares. Therefore, the fraction that is striped is \(\frac{10}{36}=\frac{5}{18}\). Two barn quilt designs using four colours and the same fractions are given below.