7139
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2011-02-01T00:00:01
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<p>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 <span style="font-weight: bold;">tenth</span> pattern in the sequence?<br></br>
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<mdo:image alt="" height="335" src="45%202010%20a1.PNG" width="820"></mdo:image><br></br>
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If you liked this problem, <a href="http://nrich.maths.org/2290&part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas.<br></br>
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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 <span style="font-style: italic;">n</span>th 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.<br></br>
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<mdo:image width="734" height="268" alt="" src="45%202010%20b.PNG"></mdo:image><br></br>
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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 <span style="font-style: italic;">n</span>th 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 $(n+4)^2 - n^2 -4 =
8n +12$.<br></br>
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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?
Sequences, Functions and Graphs
Arithmetic sequence
Sequences, Functions and Graphs
Sequences