Puzzle from
one of the greatest puzzlers of the last
two centuries perhaps, just putting it
out there; this guy called Sam Loyd. You
can look him up. This is one of his most
famous puzzles and he patented it and
sold millions of, sort of, physical copies.
So I've made myself a physical copy; it's
quite hard to get the originals and if
you do get an original, worth a lot of
money. The puzzle's called 'Get off the
Earth'. The the centre bit moves, and I
just made it with a paper clip and
printed this out. But the racial
stereotyping I want to move away
from. So there are some equivalent
puzzles which I'm going to show you, I
made the digital copy. And I found one
that Sam Loyd also made to do with a
bunch of bicyclists, they're boys with flags
around a bicycle and it moves in a
similar way. I invite you to count the
number of cyclists around this wheel. - (Brady: 1, )
(2, 3, 4, 5, 6, 7, 8, 9, )
(10, 11, 12, 13.)
Not rocket science?
On card I'd just be spinning the
inner wheel; and the same question now
Brady, count them. Where do you want to start?
(The B. 1, 2, 3, 4,)
(5, 6, 7, 8, 9, 10, 11,)
(12!)
I can see his face.
The text underneath says 'Turn the disc
so the arrow points to A and count 13
boys, then move the arrow to B and there
are only 12 boys'. So they agree with what
you've just said and the question is,
which boy has vanished and where did he
go?
Let's just check.
(Count heads.) - Okay we'll count heads: 1,
2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13. Definitely 13 heads.
Shift it around;
heads yeah? 1, 2,
3, 4, 5, 6, 7, 8, 9, 10,
11,
12
heads.
So you wanna count legs? - (I don't )
(it's- the situation's gonna change. I)
(still think it's something to do with)
(the way the boys are drawn but-) - Something
to do with the way the boys are drawn?
Do you want to say more about your
instincts here? You've got good instincts
today Brady, your instincts are
are important. So what is it about the
way they're drawn that you think you're
suspicious about? - (I just think- I just)
(think something that was a head turned)
(into not a head.)
(Like a shoe or something.) - Really? Yeah..
I mean let's find a candidate right? So
this head get- gets cut in half
but then there's a bit of a head here
that it kind of joins with, right? It
joins up there.
And it got- replaced the other half of
the head here and this one gets lost a
bit, but they're still there.
Some funky things- you know, this leg here
I think is a left leg, and when I slide
it's now a right leg. That's interesting,
I don't know if that messes up your
count but it shouldn't change how many
heads there are in my opinion. - (The key is)
(there where the two are together.) - What, here?
(That's gotta be key.)
Well, but there's still two heads right?
And they're just separated, there they
are they're still there, there's nothing
tricky going on.
(But it's- that's- I think that's the key.)
This is one of those puzzles that's
frustrating, and it caused a stir back in
the day because it feels so simple,
there's nothing tricky going; it's a
piece of paper, you can actually touch it,
you can move it around, there's nothing
to conceal really.
But something surprising is going on, you
start to doubt your counting ability,
you start counting from different places;
you count the heads, maybe you count the
flags. And it's one of those puzzles that
feels insulting to your intelligence and
that's why it splits people into like
really annoying,
something tricky is going on, and
fundamentally doubting the very core of
your being - like I can't count to 12??
Nothing should change when you spin that
so it- there's something subtle going on
and I thought I'd explain it. There's
some other variations of this puzzle, I
thought I'd show you. So Martin Gardner
wrote about this one and I found one
that he made. This time it's not in a
circle, let's do some counting together
Brady. How many faces? - (Six.)
I agree. How many faces? - (Five...)
So which face disappeared?
(I mean there are still six hats.) - And I
can shift back.
(Ahh but there's only five chins.) - Well the
chin's just a horizontal one there.
The first thing to comment is that, like
a magic trick - this is done less as a
magic trick more of just like, what's going
on? But that's what magic really is.
The question of how many faces is a
valid question, but it has two different
answers in the two configurations in the
spinny version or this version. The
second question which is, where did the
bicyclist go? Which one disappeared? Or
which face disappeared or which face
appeared,
that's the misdirection.
I think that question is
invalid
and that's why you should ask it to sort
of apply misdirection. Because if you
ask someone to describe what happens to
the faces you get a different bit of
attention. Tell me about these faces,
maybe compared to those faces; maybe tell
me about the height of them. - (They're)
(different heights.) - They're different heights,
yeah. In fact they're different faces,
none of the original faces are still
there. There's six faces, none of those
faces exist anymore, there are five new
faces. So that's why that question is
misleading I think. Which face disappeared?
All of them. And we made five new faces
which are fundamentally different, in
fact they're a bit longer.
Do you begin to see what's going on? When
you simplify it this much it begins to
get insulting. 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11 - agreed?
(Yeah all right.) - So in the other puzzles I
shifted something, I'm just gonna shift
something on this diagram.
Now I know you saw what happened, but how
many lines are there? 1, 2, 3, 4,
5, 6, 7, 8, 9, 10. And
there were 11 just now. I'm going to go back
again,
11 lines, which line appeared, disappeared,
is a nonsense question agreed? - (I don't)
(really know, I'm kind of baffled.) - This is
so simple that it's baffling. There are
definitely 11 lines here. And what I've
actually done is I sliced them along
that line and I just shifted everything
along the line, slid it there. And now
there are only 10 lines but they're all
a bit longer. So which line disappeared?
None of them. That's the effect we're
happening, when you say which line
disappeared, well it's a nonsense question.
Actually I've just made 10 new lines
from bits of the old ones that are a bit
longer.
So imagine that situation, put the lines
at different heights and slice
horizontally - so here's just
another version of the same thing. You
count the lines but then you shift them;
different number of lines.
Some of them are different lengths. I've
done this randomly and now it's really
obvious that some of them have
fundamentally changed, it's just a
different set of lines with a different
number going on. And that wouldn't be
impressive to anybody. What you could do
though, let's work this around a circle,
like the first two things we saw were
done on a circle. So it's basically
taking that line that I've sliced some
lines and wrapping it around. You can see
that I get a rotational thing here. So
let's just count the lines here: 1, 2,
3, 4, 5, 6, 7, 8, 9, 10.
And they're all a certain length. If I then
switch it round
we have a little bit of a conflict. You
can see there are two lines next to each
other, they could be counted as one line
but if they were more identifiable you
could see that they're they're two
different lines. There's two lines of
roughly that length: 1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11. If
you start getting clever wrapping this
cut around a circle, and realising all
that's happening is you're making new
lines appear which are slightly longer;
and if they're not a lot longer they
look very like the old lines, you get a
little glimpse of what an artistic
version of this could look like. So we'll
go back to the face. Imagine these are lines,
there are six lines. And if I shift them
over, now there are only five of them but
they're all a bit longer. And so if you
design the faces cleverly, this guy is-
his eyes are there, they look a bit like
eyebrows, but they actually become
eyebrows on this guy's face. So there's
some artistic cleverness to make the
identifiers a little bit subtle. And now
there are five faces where there
previously were six but they're
different faces, they're all a bit longer,
it's exactly equivalent to the lines. If
we go back to our bicyclists we've gone
back to the circle idea. We've drawn 13
boys, and you can see they're getting
they're getting further and further in
on the circle, a bit like our lines got
shifted. And when you shift it we're just
splicing together bits of boys; all the
boys disappeared and 12 new boys
appeared which are all a bit longer. But
they're designed artistically enough so
it's really hard to tell. On a bit of
paper everything's slightly imprecise
and artistically we forgive any sort of
slight misalignments, and there are quite
a few. - (So tell me with the boys,)
(where did the 12 to 13 heads change?)
(Because heads aren't lines.) - Heads aren't
lines, but there are definitely
13 heads. Except some of them are very
clear heads and obviously they're not
changing, this one up here is not
changing
and the ones inside are not changing; but
the crucial thing is around these edges
where the head is intersected by the
line.
That's where we splice together bits of
two other heads
and the new head we get is slightly
longer, or slightly shorter, or just
slightly smaller. And if you- like you can
see that this this guy's head here
is quite fat,
and when I shift it
it's even fatter,
like a super fat head. And that's
happening in a subtle way all around the
body parts, the legs, the heads - this
is a really lovely mixture of a really
simple piece of maths, like you can
splice lines and they look similar but a
bit longer,
done artistically into a way that
completely bamboozles clever people who
have their wits about them. They know how
to count 13, and 12, and yet that little
shift
changes the fundamental reality of the
number here? It's it's a nice example of
a puzzle that's been around a long time
but even when you understand it there's
still something slightly bamboozling
about it.
A great thing about the puzzle in
today's video was how it was interactive;
you could slide it back and forward, look
closely, like on your own terms. And
that's also a great thing about the
courses on Brilliant, today's episode
sponsor. These are from Brilliant's
Pre-Algebra course;
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