2018 AMC 8 Problems/Problem 19
Problem 19
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?
Solution 1
Instead of + and -, let us use 1 and 0, respectively. If we let , , , and be the values of the four cells on the bottom row, then the three cells on the next row are equal to , , and taken modulo (mod) 2 (this is exactly the same as finding , and so on). The two cells on the next row are and taken modulo (mod) 2, and lastly, the cell on the top row gets .
Thus, we are looking for the number of assignments of 0's and 1's for , , , such that , or in other words, is odd. As , this is the same as finding the number of assignments such that . Notice that, no matter what , , and are, this uniquely determines . There are ways to assign 0's and 1's arbitrarily to , , and , so the answer is .
Solution 2
The sign of the next row on the pyramid depends on previous row. There are two options for the previous row, - or +. There are three rows to the pyramid that depend on what the top row is. Therefore, the ways you can make a + on the top is , which is C.
See Also
2018 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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