2016 AMC 10A Problems/Problem 18
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[hide]Problem
Each vertex of a cube is to be labeled with an integer through
, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
Solution 1
First of all, the adjacent faces have the same sum , because
,
,
so now consider the
(the two sides which are parallel but not on same face of the cube);
they must have the same sum value too.
Now think about the extreme condition 1 and 8, if they are not sharing the same side, which means they would become endpoints of
,
we should have
, but no solution for
, contradiction.
Now we know and
must share the same side, which sum is
, the
also must have sum of
, same thing for the other two parallel sides.
Now we have parallel sides
.
thinking about
endpoints number need to have a sum of
.
It is easy to notice only
and
would work.
So if we fix one direction or
all other
parallel sides must lay in one particular direction.
or
Now, the problem is same as the problem to arrange points in a two-dimensional square. which is
=
Solution 2
Again, all faces sum to If
are the vertices next to one, then the remaining vertices are
Now it remains to test possibilities. Note that we must have
Without loss of generality, let
Does not work.
Works.
Does not work.
Works.
Does not work.
Works.
So our answer is
Solution 3
We know the sum of each face is If we look at an edge of the cube whose numbers sum to
, it must be possible to achieve the sum
in two distinct ways, looking at the two faces which contain the edge. If
and
were on the same face, it is possible to achieve the desired sum only with the numbers
and
since the values must be distinct. Similarly, if
and
were on the same face, the only way to get the sum is with
and
. This means that
and
are not on the same edge as
, or in other words they are diagonally across from it on the same face, or on the other end of the cube.
Now we look at three cases, each yielding two solutions which are reflections of each other:
1) and
are diagonally opposite
on the same face.
2)
is diagonally across the cube from
, while
is diagonally across from
on the same face.
3)
is diagonally across the cube from
, while
is diagonally across from
on the same face.
This means the answer is
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 10 Problems and Solutions |
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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 |
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