Math Dad with another challenge problem for Science Mom---are you ready? I am. Okay, this
is a geometric vanishing puzzle by Mel Stover called the Color-Changing
Pencil. Okay, so I want you to count the number of red pencils for me. ... five, six,
seven. Seven red pencils. Okay then. This figures chopped into three pieces. I'm
going to swap the bottom two pieces. Count with red pencils for me. One, two,
three, four, five, six. Six red pencils. Your challenge... What did you do? All I did was
swap the bottom two pieces; that's it. No way. All right.
Give you two minutes to come up with an explanation go. On this particular
problem, I would suggest pausing the video, clicking down the link below, and
trying this activity out for yourself. You can just quickly login to Desmos. It's
free, and there's no commitment. Log in, Try dragging this dot around and seeing
for yourself that the number of pencils of each color actually changes. It'll give
you control over the problem so you can give it a try. Can I draw on this one? Sure.
Where's my little colored things. Oh, guess you can't draw on it. Never mind. Okay no
drawing. So this pencil right here doesn't have... you know the color can be
can be changed. Hmm. what's going on. Oh, I have to actually drag okay.
Hmm, so here it's blue so I
have the pattern red, red, blue, blue, red, blue, red, blue, blue, blue. I have two Reds
over here and two go over here and then when I switch it red, red, blue,
blue, red, blue, red, blue. It's a red, blue here but it was blue, blue before.
I switched it so is it really... is it playing tricks. Alright, so when I look at
the bottom there's one, two, three, four, five, six red.
If I just look at the bottom again there's one, two, three, four, five, six, red.
There are only six red pencils. This is an illusion. Okay, now how many red do
you count? One, two, three, four, five, six. The bottom pencil with the
eraser is only part the counts. Ha ha ha ha.
Two, three, four, five, six excellent top there's only six something
weirds going on here one two three four five six there are
only six red pixels that you're asking me why does the number of pencils in
each color change there's seven red pencils apparently and now they're only
Six. Seven. Six. Seven. Six. Yeah, it's because of this guy right here.
Essentially you're correct. That is one point where you could claim
that's the crux. Things change right there when we do the swap. That one
changes---isn't gonna change-- has your red to red. Yeah, but that one doesn't have a
fixed color, or alternatively you also kind of pointed out this eraser over
here could go with either color of pencil. Yeah so when we drag it across
all of the red pencils each got a little bit longer then, and there are only six
of them. Wwhen I drag it back all the red pencils get a little bit shorter on
average. Mm-hmm. But now there's a seventh one so all of the mass was hidden
or redistributed when we swapped the two body pieces. This is the type of problem
that's guaranteed to mess with your head for a minute. Yeah the mass of the red
pencils was just redistributed, and the pictures aren't changing so you have the
same amount of red in Each picture. Right but when we go from
seven red pencils to six red pencils each of them gets a little bit
longer on average. So the red parts are redistributed but then there
are actually fewer red lengths. So there's seven blue pencils here and six
there, so they just swap back. Right so the same thing was going on with the
blue pencils as was happening with the red pencils. Okay, so how are we gonna decide
on the points of that one? I think my explanation is as good as yours; only
the erasers count. I don't know. I think you were pretty flummoxed at the
end. But if I just said it with more confidence...
No! There's definitely a red pencil up there. "There are only six red pencils".
No. There they were seven.
If you enjoyed this problem I would suggest doing a
Google search for "geometric vanishing puzzles." You're guaranteed to find some
puzzles that will really mess with your head. Lots more like this one? Lots like
this one.
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