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Difference between revisions of "1981 AHSME Problems/Problem 22"

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==Problem==
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==Problem 22==
 
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?
 
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?
  
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<br>
 
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(This was my first solution, apologies if it is bad).
 
(This was my first solution, apologies if it is bad).
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==See also==
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{{AHSME box|year=1981|num-b=21|num-a=23}}
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{{MAA Notice}}

Revision as of 13:16, 28 June 2025

Problem 22

How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four?

$\textbf{(A)}\ 60\qquad\textbf{(B)}\ 64\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 100$

Solution 1(casework)

Restating the problem, we seek all the lines that will pass through ($i$, $j$, $k$), ($i + a$, $j + b$, $k + c$), ($i + 2a$, $j + 2b$, $k + 2c$), and ($i + 3a$, $j + 3b$, $k + 3c$), such that $i,j,k$ are positive integers, $a,b,c$ 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:

  1. Set $a/b/c$ to $0$. This then allows us to set the corresponding $i,j,k$ to any number from $1$ to $4$, inclusive.
  2. Set $a/b/c$ to $1$. This forces us to set the corresponding $i/j/k$ to $1$.
  3. Set $a/b/c$ to $-1$. This forces us to set the corresponding $i/j/k$ to $4$.

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 $6$ ways to assign each coordinate, for a total of $216$. We then must subtract the ways to get a point ($4$ ways per coordinate, for a total of $64$). This leaves us with $152$. Finally, we divide by $2$ to account for mirror assignments giving us the same coordinate, for a final answer of $76$.
(This was my first solution, apologies if it is bad).

See also

1981 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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