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Problem of the Week
Problem A and Solution
Navid’s Nets

Problem

When Navid recycles boxes, he flattens them. Since he is learning about nets in school, he notices there are many different ways to flatten a cube-shaped box into a net.

A net is a pattern that can be cut out and then folded together to create a 3D shape. For example, consider the net shown.

A figure is formed using six identical squares. Four squares are placed horizontally side by side forming the centre row of the figure and the remaining squares are placed above and below the second square in the centre row.

When the net is folded together, it makes a cube.

Navid draws all the possible nets for a cube and finds that there are eleven different nets. Draw as many of the nets as you can. Can you draw them all?

Solution

The eleven possible nets for a cube are shown. Note that the nets are all considered different because for each net, flipping or rotating the net does not produce any of the other nets.

Four squares form a vertical centre column with an additional square on either side of the first square in the centre column.   Four squares form a centre column with an additional square to the right of the first square in the centre column, and an additional square to the left of the second square in the centre column.   Four squares form a centre column with an additional square to the right of the first square in the centre column, and an additional square to the left of the third square in the centre column.   Four squares form a centre column with an additional square to the right of the first square in the centre column, and an additional square to the left of the fourth square in the centre column. Four squares form a centre column with an additional square on either side of the second square in the centre column.   Four squares form a centre column with an additional square to the right of the third square in the centre column, and an additional square to the left of the second square in the centre column.   Three squares form a centre column with an additional square to the left of the second square in the centre column and an additional square to the right of the first square in the centre column with a final square on top of it.   Three squares form a centre column with an additional square to the left of the third square in the centre column and an additional square to the right of the first square in the centre column with a final square on top of it. Two columns of three squares are placed side by side so that the bottom square in the left column lines up with the top square in the right column.   Three columns of two squares are placed side by side so that the bottom square in the left column lines up with the top square in the middle column, and the bottom square in the middle column lines up with the top square in the right column.   Three squares form a horizontal centre row. Two additional squares are placed below the middle square in the centre row forming a capital T shape. Another square is placed above one of the squares on the end of the centre row.