Difference between revisions of "1981 AHSME Problems/Problem 22"
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==Problem== | ==Problem== | ||
− | How many lines in a three dimensional rectangular | + | How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form <math>(i, j, k)</math>, where <math>i</math>, <math>j</math>, and <math>k</math> are positive integers not exceeding four? |
<math>\textbf{(A)}\ 60\qquad\textbf{(B)}\ 64\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 100</math> | <math>\textbf{(A)}\ 60\qquad\textbf{(B)}\ 64\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 100</math> | ||
− | ==Solution== | + | ==Solution 1(casework)== |
− | + | Restating the problem, we seek all the lines that will pass through (<math>i</math>, <math>j</math>, <math>k</math>), (<math>i + a</math>, <math>j + b</math>, <math>k + c</math>), (<math>i + 2a</math>, <math>j + 2b</math>, <math>k + 2c</math>), and (<math>i + 3a</math>, <math>j + 3b</math>, <math>k + 3c</math>), such that <math>i,j,k</math> are positive integers, <math>a,b,c</math> are integers, and all of our points are between 1 and 4, inclusive. With this constraint in mind, we realize that for each coordinate, we have three choices: | |
+ | # Set <math>a/b/c</math> to <math>0</math>. This then allows us to set the corresponding <math>i,j,k</math> to any number from <math>1</math> to <math>4</math>, inclusive. | ||
+ | # Set <math>a/b/c</math> to <math>1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>1</math>. | ||
+ | # Set <math>a/b/c</math> to <math>-1</math>. This forces us to set the corresponding <math>i/j/k</math> to <math>4</math>. | ||
+ | Note that options 2 and 3 will give us the same points if we mirror the assignments of each coordinate. Also note that we cannot set all three coordinates to not change, as that would be a point. | ||
+ | <br> | ||
+ | All of this gives us <math>6</math> ways to assign each coordinate, for a total of <math>216</math>. We then must subtract the ways to get a point (<math>4</math> ways per coordinate, for a total of <math>64</math>). This leaves us with <math>152</math>. Finally, we divide by <math>2</math> to account for mirror assignments giving us the same coordinate, for a final answer of <math>76</math>. | ||
+ | <br> | ||
+ | (This was my first solution, apologies if it is bad). |
Latest revision as of 19:46, 9 September 2024
Problem
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form , where
,
, and
are positive integers not exceeding four?
Solution 1(casework)
Restating the problem, we seek all the lines that will pass through (,
,
), (
,
,
), (
,
,
), and (
,
,
), such that
are positive integers,
are integers, and all of our points are between 1 and 4, inclusive. With this constraint in mind, we realize that for each coordinate, we have three choices:
- Set
to
. This then allows us to set the corresponding
to any number from
to
, inclusive.
- Set
to
. This forces us to set the corresponding
to
.
- Set
to
. This forces us to set the corresponding
to
.
Note that options 2 and 3 will give us the same points if we mirror the assignments of each coordinate. Also note that we cannot set all three coordinates to not change, as that would be a point.
All of this gives us ways to assign each coordinate, for a total of
. We then must subtract the ways to get a point (
ways per coordinate, for a total of
). This leaves us with
. Finally, we divide by
to account for mirror assignments giving us the same coordinate, for a final answer of
.
(This was my first solution, apologies if it is bad).