2009 AMC 12A Problems/Problem 11
Contents
[hide]Problem
The figures , , , and shown are the first in a sequence of figures. For , is constructed from by surrounding it with a square and placing one more diamond on each side of the new square than had on each side of its outside square. For example, figure has diamonds. How many diamonds are there in figure ?
Solution
Solution 1
Color the diamond layers alternately blue and red, starting from the outside. You'll get the following pattern:
In the figure , the blue diamonds form a square, and the red diamonds form a square. Hence the total number of diamonds in is .
Solution 2
When constructing from , we add new diamonds. Let be the number of diamonds in . We now know that and .
Hence .
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 |