Doubly Symmetric
What is the smallest number of additional squares that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2?
Age
11 to 14
Challenge level
Problem
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What is the smallest number of additional squares that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order $2$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
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The shape must have rotational symmetry of order 2, so when it is rotated through 180$^\circ$, the squares that the shaded squares rotate to must also be shaded.
These are marked in red on the diagram.
This pattern has two lines of symmetry, along the diagonals of the square, so it satisfies both criteria.
Therefore the minimum number of extra squares that need to be shaded is 3.