Square holed pattern
Weekly Problem 45 - 2010
This pattern is made from small shaded squares. Can you picture where the patterns lead? How many squares will you need for the tenth pattern?
This pattern is made from small shaded squares. Can you picture where the patterns lead? How many squares will you need for the tenth pattern?
Age
11 to 14
Challenge level
Problem
The diagram shows the first three patterns in a sequence in which each pattern has a square hole in the middle. How many small shaded squares are needed to build the tenth pattern in the sequence?
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If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
One way to proceed is to regard the pattern as four arms, each two squares wide, with four corner pieces of three squares each. So for the nth pattern, we have $4\times 2\times n + 4\times 3 = 8n+12$. For $n = 10$, we need $ 8 \times 10 +12$ i.e. $92$ squares.
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Alternatively, it is possible to see the patter as a complete square with corners and a central square removed. So for the nth pattern, we have a complete $(n+4)(n+4)$ square with the four corners and a central $n\times n$ square removed. Hence the number of squares is $(n+4)^2 - n^2 -4 = 8n +12$.
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