Difference between revisions of "2018 AMC 8 Problems/Problem 19"
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~cxsmi (Note) | ~cxsmi (Note) | ||
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+ | ==Sulotion 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 <math>+</math> on the above, half of them have a <math>-</math> above. So, For the lowest layer with <math>4</math> blocks, there are <math>2^4=16</math> possible combination and half of them will lead a <math>+</math> (or <math>-</math>) on the top. The answer is <math>16/2=\boxed{\textbf{(C) } 8}</math>. | ||
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+ | If you notice this rule, you can give the answer whatever how many layers you have. The answer will be <math>2^{n-1}</math> for the layer with <math>n</math> blocks. | ||
==Video Solution (CREATIVE ANALYSIS!!!)== | ==Video Solution (CREATIVE ANALYSIS!!!)== |
Revision as of 04:57, 5 March 2024
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)
Sulotion 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.
Video Solution (CREATIVE ANALYSIS!!!)
~Education, the Study of Everything
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 | |
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 AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.