G
remmie’s
F
un
P
ages
edited by Gerald Muus
Solution to Last Issue’s Brain Teaser:
84
92
80
?
You may recall that we needed to know the correct number to replace the question mark. We asked that you provide your
methodology as part of the solution.
Although it is possible to arrive at the answer by virtue of trial and error, we were looking for a scientific answer. We were not
disappointed. We received a torrent of replies, most of which contained what we wanted: an algebraic solution.
The first step is recognizing that we have three variables. We may choose to represent them as a, b, and c, or x, y, and z.
Let (star) = x
Let (circle) = y
Let (triangle) = z
From the diagram, we can construct three equations:
x + 2y + z = 84
2
x + 2z = 80
2
x + y + z = 92
Let’s look at the middle of these:
2
x + 2z = 80
x + z = 40
x = 40 – z
Now if we substitute (40 – z) for x in the first equation, we get:
(40 –
z) + 2y + z = 84
40
+ (-z) + 2y + z = 84 and therefore we eliminate z
40
+2y = 84
20
+ y = 42
y = 22
Now let’s substitute (40 - z) for x and the known value of y into the last equation:
2(40 –
z) + 22 + z = 92
80
+ (-2z) + 22 + z = 92
80
+ 22 + z – 2z = 92
80
+ 22 –z = 92
102 –
z = 92
102
= 92 + z
z = 102 – 92
z = 10
Since we know that x = 40 – z, we find that x = 30.
We may now write a fourth equation thusly, to arrive at the answer:
x + y + 2z = ?
Plugging in the numbers, we see that:
30
+ 22 + 2(10) = ?
30
+ 22 + 20 + ?
52
+ 20 = ?
?
= 72
Thanks to the huge number of readers who solved this puzzle correctly. The first was Kathleen Tully of Union Hall, followed by
many others from around the lake, and a few from out of state visitors.
As Assistant Editor of
Discover Smith Mountain
Lake, “Gremmie” loves
to get mail. Here she is,
busy at work, looking over
the entries in our puzzle
competition.
If you care to drop her a
line, send it to:
Gremlin the Wonder Cat
Assistant Editor
Discover Smith Mountain
Lake Magazine
P.O. Box 880
Moneta, VA 24121
Or via e-mail at:
gremmie@discoversmith-
mountainlake.com
Discover Smith Mountain Lake |
www.discoversmithmountainlake.com
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Discover Smith Mountain Lake | Fall 2012
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