2018 AMC 8 Problems/Problem 19
Contents
Problem
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
You could just make out all of the patterns that make the top positive. In this case, you would have the following patterns:
+−−+, −++−, −−−−, ++++, −+−+, +−+−, ++−−, −−++. There are 8 patterns and so the answer is .
-NinjaBoi2000
Solution 2
The top box is fixed by the problem.
Choose the left 3 bottom-row boxes freely. There are ways.
Then the left 2 boxes on the row above are determined.
Then the left 1 box on the row above that is determined
Then the right 1 box on that row is determined.
Then the right 1 box on the row below is determined.
Then the right 1 box on the bottom row is determined, completing the diagram.
So the answer is .
~BraveCobra22aops
Solution 3
Let the plus sign represent 1 and the negative sign represent -1.
The four numbers on the bottom are , , , and , which are either 1 or -1.
Which means = 1. Since and are either 1 or -1, and . This shows that = 1.
Therefore either , , , and are all positive or negative, or 2 are positive and 2 are negative.
There are 2 ways where , , , and are 1 (1, 1, 1, 1) and (-1, -1, -1, -1)
There are 6 ways where 2 variables are positive and 2 are negative: (1, 1, -1, -1), (1, -1, 1, -1), (-1, 1, 1, -1), (-1, -1, 1, 1), (-1, 1, -1, 1), and (-1, -1, 1, 1).
So the answer is .
Note: This result can also be achieved by realizing that there are ways to arrange negatives and positives and way each to arrange four of one sign.
~atharvd
~cxsmi (Note)
Solution 4
The pyramid is built on the basic 3 blocks pattern: one above and two below. The basic pattern have four possible symbols and half of them have a on the above, half of them have a above. So, For the lowest layer with blocks, there are possible combination and half of them will lead a (or ) on the top. The answer is .
If you notice this rule, you can give the answer whatever how many layers you have. The answer will be for the layer with blocks.
Solution 5
We can use casework to solve this problem. The only way for the top cell to have a in it is if the third row of the pyramid (the one with cells) is either or . First, let's pretend that the third row of the pyramid is . The only way for that to happen is if the second row (the one with cells) is --- or . Now, let's pretend that the second is . That would have possibilities for the first row (the one with cells), and . Next, let's pretend that the second row is ---. That makes two more possibilities for the first row, and . Now, let's pretend that the 3rd row is , which means that the second row is either or . You will soon find that find possibilities for the first row, or , and possibilities for , and . Together, we find that the answer is $
Video Solution (CREATIVE ANALYSIS!!!)
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Video Solution
~savannahsolver
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s
See Also
2018 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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