Difference between revisions of "2000 AMC 12 Problems/Problem 8"
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Note that figure 0 has 1 square, figure 1 has 5 squares, figure 2 has 13 squares, and so on. If we let the number of the figure = <math>N</math>, note that <math>N^2 + (N+1)^2</math> represents the number of squares in the figure. For example, figure 4 has <math>4^2+5^2 = 41</math> squares. Therefore, the number of squares in figure 100 has <math>100^2 + 101^2 = 20201 \Rightarrow\mathrm{(C)}</math>. | Note that figure 0 has 1 square, figure 1 has 5 squares, figure 2 has 13 squares, and so on. If we let the number of the figure = <math>N</math>, note that <math>N^2 + (N+1)^2</math> represents the number of squares in the figure. For example, figure 4 has <math>4^2+5^2 = 41</math> squares. Therefore, the number of squares in figure 100 has <math>100^2 + 101^2 = 20201 \Rightarrow\mathrm{(C)}</math>. | ||
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+ | <math>2^{\text{nd}}</math> alternate solution: | ||
+ | For the <math>n^{\text{th}}</math> figure, note that it could be constructed by making a <math>(2n+1)\times (2n+1)</math> square, and then removing the <math>n^{\text{th}}</math> triangular number from each of its corners. So, if <math>a_n</math> represents the amount of squares in figure <math>n</math>, <math>a_{n} = (2n+1)^2-4\frac{(n)(n+1)}{2}= 2n^2+2n+1</math>. Therefore, <math>a_{100} = 20201</math>, which gives <math>\mathrm{C}</math>. | ||
== See also == | == See also == |
Revision as of 00:44, 3 February 2009
Problem
Figures ,
,
, and
consist of
,
,
, and
non-overlapping squares. If the pattern continued, how many non-overlapping squares would there be in figure
?
Solution
By counting the squares starting from the center of each figure, the figure 0 has 1 square, the figure 1 has squares, figure 2 has
squares, and so on. Figure 100 would have
.
Alternate solution:
Note that figure 0 has 1 square, figure 1 has 5 squares, figure 2 has 13 squares, and so on. If we let the number of the figure = , note that
represents the number of squares in the figure. For example, figure 4 has
squares. Therefore, the number of squares in figure 100 has
.
alternate solution:
For the
figure, note that it could be constructed by making a
square, and then removing the
triangular number from each of its corners. So, if
represents the amount of squares in figure
,
. Therefore,
, which gives
.
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |