Here's a favourite part of my collection - the
Missing Area Paradox I call these. This is Sam Lloyd's 1895 invention... wonderful! Variously
called Vanishing Area and Redistributed Area, and I'll explain why it's called that. This
came out in 1895 - this is a reproduction - and it's magical. So, what is the magic? Well,
I need to put it on the table to show you. It's called 'Get Off The World' puzzle, much
loved by the Americans; and it shows a number of warriors all the way around it; we start at NE
for North East, and count the number of people we've got. One two three four five six seven eight
nine ten eleven twelve thirteen people we've got. What we do is we push this round to the North
West, so that's now lined up with North West, and do a re-count. Here we go... one two three
four five six seven eight nine ten eleven only 12... where's the 13th man gone? Oh! When
you turn it like that, it's back to 13; where has he come back from? So you can't tell, can
you? If you turn it a bit more, you get some strange distortions, and you'll start to realize
that what's actually happening is you've got 13 slightly smaller people, or 12 a little bit
larger; each of them has borrowed a little bit of each other's bodies; the total area of the
figures is the same, but it's so cleverly drawn you would never notice that. So a brilliant start.
I'll show you a couple more of this genre, because I think they are every bit as enjoyable to show
people. This one for instance is probably about 10 or 20 years old, so it's quite modern. It's
showing a nice blue bird, who's busily hatching things... hatching green eggs; well, laying green
eggs and then they are hatching. Here we go, let's do a count first of all. We'll start here
counting eggs - one two three four five six seven eight eggs; and the chicks that have already been
born are... one two three four five six seven; what happens when you move the arrow
pointing to 'A' to over here is... what you expect... knock knock knock it goes, and
breaks open, and one of the chicks is born, and the egg turns into a chick. Let's see if that's
what happens shall we? Starting here with eggs... one two three four five six seven eggs only, there
were eight; oh! And the chicks are one two three four five six seven eight chicks... yes, one of
the eggs has hatched, and turned into a chick; what good news for mum! Isn't that clever! And
there's one final one to show of this genre, which is this one here, because it's so beautifully
drawn, showing galleons or sailing ships; this arrow here it's going to point up to the top
- it's going to be this way round - to 12 o'clock. And how many galleons are there? Well you can tell
easily, because they're matching the hours of the clock, so there's going to be 12 of them. Notice
that they have put XIII or 13 instead of one. So there's 12 there, and if we point arrow to the
one o'clock position, or the 13 up here, we'll find that an extra one appears, which they call a
'pirate ghost galleon', and sure enough - one two three four five six seven eight nine ten eleven
twelve thirteen - yes, an extra one has appeared; and you can see it's the same idea, but with
a bit of romantic history there I think. And now we come to a completely different
approach to this business of missing area paradoxes... it's the slider one, which I
think is sometimes a little more subtle. Here's just three examples of it, starting with one
that was sent to me as a Christmas card actually, by Richard Robinson back in 1999; and it's
showing Santa Claus here with a problem, because he's got ten chimneys here, no nine it
should be... one two three four five six seven eight nine chimneys; count them up - we've got one two
three four five six seven eight nine ten stockings... oh... one of them has got to drop on the ground because
there's no chimney for it; how to solve this problem? Poor old Santa is getting quite upset as
you can see. Well all you do is slide... you just slide from there to there, that's
all you need to do, and now we find to our joy, the chimneys stay the same of course, but now
we've got one two three four five six seven eight nine stockings for nine chimneys and we've got
a happy Santa Claus riding away into the sky. Isn't that a nice way of presenting the same thing -
each of these has got a little bit longer look... that's that bit there, and now it's a little
bit shorter, it lost its top, and so on; and that's a bit of a top end that had a tiny bit
of the very bottom there, that's disappeared; so we've got some slightly larger but fewer
stockings, and slightly smaller but more stockings; and here's another version of the same idea which
is a little more elaborate, but I think it's fun because it shows when you take it a stage further
how ridiculous the actual artwork becomes; we've got a series of galleons here, no, sailing ships, and
we've got one two three four five six seven eight nine ten sailing ships, and all you do is slide it
down to the next position, and we find we've got one two three four five six seven eight
nine ten eleven, that's right eleven, because I'm gonna now do that, I'm gonna lose the flag at the
top and he gets to 12; You can actually make it 10 11 or 12, depending
on where you put it, there or there or there; but there's a flag that appears; so this is quite
clever because you can make it quite subtle. If you go any further you get ridiculous situations where
the boats here don't have any hulls for instance, which is ridiculous; but the three best
positions really are where you've got 10 11 or 12 sailing ships, all done by sliding
this thing to and fro; very nicely done. And the last one is something that's very new
and absolutely ingenious, because it helps to show what's going on. This is something that Martin
Gardner drew up about 30 years ago; he made these little figures here, and suggested
you should slide the pieces, and we'll see if this person who is invisible becomes visible; we've got
one two three four five people, but here's a clever bit which someone has recently invented, which is
to give them all names; so here we've got Joey and Ronald and Donnie and Robert and Allen; when we move it
like that to there, now we've got one extra person one two three four five six...
everyone's under a hat now, and look what's happened to the names - we've got Joe we've got
Roy we've got Donald we've got Ronnie we've got Albert and we've got Len; we've got extra
people appear, and of course all you've done is redistributed the letters to have
five slightly longer words and six slightly shorter words, that's all it is; it's
a nice way of demonstrating what's going on. So that's a brilliant way of finishing that line.
Now the next chapter of this is this lot here; This is something where you
don't slide the pieces, you just move pieces around, and
the simplest version is this big one here, you've just got three pieces - most
of them consist of three pieces - and you start with trolls; this is a Swedish
designer who I've corresponded with, Harry Lange from Sweden, and he did this many
years ago; it shows one two three four trolls... when you swap the pieces around you're going
to cut here and cut here and then just swap them around, so that's going to go there and
that's going to go there from that position to that position, something magic happens when
you match them up because now we find we've got only three trolls - one of them has disappeared and
where has he gone? And when he comes back again, where's he come from. Well, look at this bit here...
that looks like some of the landscape, doesn't it? Is it the top of a rock? Watch what happens
when I put them back to the other position... you'll find it actually becomes the top of his
head... clever clever clever bit of artwork there; and he disappears most of his face behind the
flowers, which is fair enough; but you can sort of believe there are four trolls there, but of course
you can see quite clearly now there are four rather smaller heads, and the other way is
three but rather larger heads - beautiful! And there's two other versions of this which I
need to show you; this one is I think one that Mel Stover produced quite a few years ago, and it's
particularly nice because you've got things going two ways. Let's see if I can get them in the right
order, yes there we are; that's the order here... We've got a problem in the local pub, because we've
got some drinkers, very thirsty, and they all want a drink, but they can't have it, because look - we've
got one two three four five six thirsty guys six blokes all with different hats, and only
one two three four glasses of beer; there's going to be a riot, there's going to be
a riot! So put a little magic in, and swap these pieces around, and see what that does. Put that
there and put that there, and by magic we've now got one two three four five blokes, thirsty
people, I don't know where the sixth one's gone; but we've got one two three four five glasses of
beer - oh joy, peace and quiet, because they've all got their glass of beer there won't be a riot
after all! Oh well. The last to show in this genre of sliding pieces I think is my favourite, because
it's a lovely depiction of Alice in Wonderland, and this time she's looking up at the
Cheshire cats - it's actually a family which is rather fun; so there's all the Cheshire
cats up in the tree, and there's one two three four five Cheshire cats, all in the family, looking down
at Alice here. Swap the pieces around and something extraordinary happens, which is what Lewis
Carroll mentioned in his book; we have one two here, and then we will put this piece together, and by
magic one of them has disappeared but not completely disappeared, because a smile is still
there... look at that... it's turned into a smile; that was a chin of the previous one and that was
the top of the head of the previous one as well in the previous position; very cleverly done so...
that's delightful I think. A very nice version of the changing the pieces around/missing area paradox;
so there's a couple more genres to show you; Just to mention the history of these which I think
is important, because the sliding ones particularly reflect this; these are very old precious documents,
I've got to been very careful with these; very very old; I know its age, because it's dated;
this actually shows those sliding pieces I was just doing, and it's 1774.
This goes back 250 years.. astonishing isn't it! That's how far back in time it goes; I was
fascinated to find that; so a bit of history. And here's one, before we come to the last
section, which is peculiar; It's a Japanese designer - a friend of mine
called Peter Hayek made a lovely version of this; It's called the ill-fated school trip... in this
case you've got to swap three pieces around, not two pieces; that's the final position,
but look at this... one two three four five six seven eight young kids,
dashing about the mountains having a lovely time; if we take the pieces out and do a little bit of
redistribution, that piece is going to go up to there, this comes down to here, and then we put
the other pieces together, and see what's happened to the kids; and there's going to be a calamity,
there's going to be shouts of dismay and everything else,
because we find that there's going to be a gap somewhere here... oh goodness me
yes, look at this... this is tragic... I'll just line that up... that's right...
we've got one two three four five six seven children only, and a space... and they show
it here as a cross - someone's disappeared, and died? Or disappeared into the wide blue yonder?
We don't know. But that's a very clever way of of producing that type of illusion where you move
the pieces around; and then Sam Lloyd was involved in something else, which again he made very very
popular, which involved a puzzle where you had this situation - two horses back to back, and two
riders, and you had somehow to put them so the two riders were sitting on the horses normally; if
you do the obvious thing which is put that across to there, you'll find the riders are attempting
to ride upside down and facing the back, so that's completely impossible; neither of those fits.
It's ridiculous. What you do is very counter-intuitive... If you place them like that, and suddenly as if
by magic the two horses have broken into a big gallop, with their fore legs
and their back legs stretched out, doing a very fierce gallop; and the cowboys are
having a whale of a time, sitting and playing their banjos. Isn't that a clever idea! Quite a
number of these were produced over the years and the most unusual and different that I've ever
come across is this one here, which is totally bizarre; it's very simply drawn,
but it's just a pair of hands stroking something, and a pair of cats; and it's no
good doing that, because they're trying to stroke the underbelly of the cat,
and the cat's not lying down; you've got to do this, like this, and now you've
got the two hands reaching down, and gently stroking the backs of the
cats which is much more realistic. Very nice indeed. So it's been a wonderful thing
to collect this, and I'm just so pleased to be able to have so many different versions of it; and
just recently, and literally it's only happened in the last few months, I've come across one of
the earliest versions which is the rotating one I began with, but in three dimensions! And
it's this wonderful piece - here look at this! It's the pièce de résistance they call it!
It's an ashtray, and it's cigars on an ashtray, we're going to do a little count,
starting with the one with the label on it. One two three four five six seven, okay? And we find
the middle bit will actually turn round; if I go one two and three... what's happening now?
We'll start with this one again... one two three four five six
seven eight... one two three four five six seven eight! Oh my goodness
me! Let's do two more turns and see what happens; and now we've got
one two three four five six... That's extraordinary... six seven eight... eight seven
six... you just can't believe it's happening, and it's all in three dimensions too;
and the cigars have all got tops and bottoms, and ash and things... so a wonderful
piece of sculpture this, quite the best version I've ever come across; and this has to be
probably my best toy in the last five years I think... Wonderful!!
