---
title: 'The Unsolved Lollipop Problem - Numberphile'
source: 'https://youtube.com/watch?v=v8e-tYey7ts'
video_id: 'v8e-tYey7ts'
date: 2026-06-30
duration_sec: 837
---

# The Unsolved Lollipop Problem - Numberphile

> Source: [The Unsolved Lollipop Problem - Numberphile](https://youtube.com/watch?v=v8e-tYey7ts)

## Summary



## Transcript

What's a lollipop? Tell me about
lollipops. Yeah, well, this is a
mathematical lollipop. What it consists
of
it is a circle and a
on a stick.
And if you continued the stick, it would
pass through the center of the circle.
This part of the stick is invisible and
the stick is infinite. So, it's a a
circle with a perpendicular stick coming
out of it, perpendicular to the edge of
the circle. That's a lollipop. And we've
got one of them and it looks like that
and it divides up the region. If it's If
we're doing them in the sand, it divides
up the beach into the part that's inside
the circle and the rest of the beach.
And so, one lollipop divides the beach,
the paper, into two regions, inside the
circle and all the rest. Because this
goes off to infinity. What if we have
two lollipops? This is actually a little
tricky. If you have two lollipops, how
many regions can you make by overlapping
them? Well, you could do this. You could
put your other lollipop here and its
stick could go here. And then, if we did
that, how many regions would we get?
One, two, three, four, five, six. We get
We could easily get six regions.
>> not get seven? Does that If these go
forever, is the other side of the thing
not a seventh? It's seven.
>> It is seven, right.
>> Yeah, yeah, you're right. Yeah, there
you go. I know my stuff. You know your
stuff.
>> Seven. All right. So, each stick goes
off to infinity. So, with with two, we
can certainly get seven. And that's
okay, but it's not great. You can
actually get 10 pieces if you do it
right. And what you do is you make sure
that the circles of the
lollipops overlap in just a little
sliver. And then, you make sure the
stick of the first lollipop cuts the
circle of the second lollipop. Okay,
yeah. Yeah, and then, scalpel, blue,
we get
this. It's going to go through the the
center imaginary through the center of
that. It's going to cut across this gulf
where the two circles
and then it's going to cut that one. So,
I'll give it a little bit of space.
Looks like that. And what you get let's
count the pieces. How many pieces do we
get when we do that? It's very careful.
We all right. We have first of all, we
have below the the sticks and above the
sticks. They they two infinite regions.
All right, one, two, then three and
four. Yeah. And then five for the
sliver, six and seven down here.
>> Yeah.
And then eight, nine, 10. So, we get 10
pieces. You can show using higher
mathematics Euler's formula that the
crucial thing
if you want to get the most pieces
what you need to focus on are the
intersections, the crossings between one
lollipop and the other lollipop. And the
crossings it's it's crucial that you
notice. You using Euler's formula you
can show that the important thing is to
get the most intersections between the
lines of one lollipop and the lines of
the other lollipop. And the
intersections
the there are three kinds of
intersections. There are intersections
where the circle part cross and you can
see I've managed to make this circle and
this circle cross in two points.
There's also
intersections where the stick of one
lollipop crosses
the circle of the other lollipop and
we'd like each of them to happen twice.
So, we can see that blue stick crosses
the other lollipop twice. Correct. And
vice versa, this the stick crosses this
one here and here. And then the sticks
themselves can cross.
And they do here at just once. Sticks
either cross or they don't. So, with two
two circles, you get seven
intersections.
And there's a formula, the number of
pieces equals intersections number of
intersections plus n, the number of
sticks plus one.
With two, n equals two,
we got seven intersections. I showed you
2 + 2 + 2 + 1, seven, and then two
because we got two circles plus one, and
that's 10. And that's the best we can do
with two lollipops. And if we had three
lollipops, ideally, we'd make each pair
of lollipops intersect in this way. And
it looks like this. All right. It is
very tricky to draw. Okay, so the new
lollipop is the green one. That's
>> one at the bottom. Okay.
>> Yeah. And it intersects the red one in
the same way that the blue and the red
intersected, and it intersects the blue
one in the same way. Each pair of
lollipops here
meet in seven intersections. And the
stick from the third new lollipop Yeah,
it's going up. But it's slightly
off-center, so that it doesn't it
doesn't intersect with the intersection
of the first two sticks. Correct.
>> To maximize our sections.
>> Yeah, you never want to have three
things meeting at a point because you
make a tiny little change and you pick
up one piece, one region. So, that So,
that stick is slightly off Slightly
off-center, and it's true. You might
think I'm fudging this, but actually, if
you work to multiple precision and you
draw it carefully, I did actually draw
it carefully.
And you can see it in the OEIS entry for
this sequence. So, the So, how many
pieces did we end up with for three
lollipops? We need to know how many
intersections there are. Each pair
intersect in seven points. So, there are
seven intersections there, seven there,
and seven there. And none of them have
been counted twice.
>> No, they're all distinct. You can see
check I'm very careful not to have any
triple points or higher. So, we got 21
intersections. All right, n equals
three. All right. Three lollipops. The
number of intersections is equal to 7 +
7 + 7 each because each we got three
lollipops and each pair meets in seven
points.
>> Yeah. And they're all different. So,
that's 21 and then the formula is this.
We add n, which is three, and we add
one, we get 25. So, with three
lollipops, we get 25 regions. But, where
are we going to put the fourth lollipop?
>> That's all I can think of.
>> This is
really, really hard. Yeah.
Cuz you cuz if you put it up You put it
up there, it's not going to meet
>> No. No.
>> Yeah. No, it is really hard. Where does
the fourth lollipop go?
>> Where does the And I tried and I did
various drawings. And we know what we
want. We want the maximum number of of
intersections between all the pairs of
lollipops. With with four lollipops,
we've got six intersections. So,
ideally, we'd get 6 * 7 42
intersections.
Let's give people some thinking time.
Yeah.
>> All right. What's the answer? Well, it
what didn't come very easily. On
Christmas Eve, I posted a message to the
Secret Santa mailing list explaining
this problem and asking for help. It
said it for I said with four lollipops,
it's really tricky. And at 1 minute past
midnight, I got an email from a couple
of old friends who said that they could
get
43 regions. The maximum would be 47.
If you could get every pair to meet in
seven points, you'd get 42 + 4 + 1,
you'd get 47 regions. They got close,
but
but not not very close. And how And how
did they uh place their circles to do
that? Well, they took my drawing of
three circles, and they added a fourth
circle, which they got by perturbing one
of the three a little bit. So, we still
crossed most of the things the same way.
>> They didn't perturb one of the existing
circles. They They added
>> They They put their new lollipop on top
of the existing lollipop and perturbed
that one. Yes. They took a copy of the
red one and perturbed it a bit. Maybe
they changed the the the diameter a
little bit, I'm not sure. And they
changed the angle of the stick. And that
gave them 43. And that gave them 43
regions.
Cool.
It was pretty good. Yeah. And for the
first 12 hours, that was the world
record. And then
2 minutes past noon on Christmas Day, I
got an email from
someone on the Sequence Fans mailing
list who I've never met, although since
we've talked on Zoom.
He was able to get 44 regions. But
later, he got it up to 45. 45 regions.
And what's more, he proved that was
optimal. So, there was no point in
anyone trying to get more. You could
theoretically have gotten 46 or 47, but
you can't. He proved that 45 is best
possible. Yeah. What he did was really
extraordinary.
He took those three, and he magnified He
modified them a little bit. He made one
rather bigger than the other two, about
twice as big. And then he blew blew it
up, magnified it by a factor of 100. So,
these circles got really, really huge.
And when you looked at the edge of the
circle, it was the circle was so huge,
it the edge looked almost like a
straight line.
And I will show you what those straight
lines looked like.
And here's a picture of how it looks
after he's blown it up. That green line
is part of a gigantic circle. Yeah, and
that's the stick of the green lollipop.
>> That's the stick of the green line.
And the red [clears throat]
>> And the red is also that's
>> The red is the
It's this
red circle magnified so that that arc
looks like a straight line. And there it
is. And that's the stick. And then blue,
this is the blue circle. Yeah. And
here's the blue stick. So he he
magnified it. And then he very cleverly
put a fourth circle on top of the place
where the sticks come together. So that
little black lollipop, that's the new
lollipop and it's miniaturized right in
the mess between the other three.
Exactly. Yes, brilliant. So that So
there it is. And if we go back and look
at the previous picture,
the extra fourth lollipop, the black
lollipop is actually here. You just
can't see it. It's so tiny. It's in
blots of ink where the three circles
and the three sticks come together.
Neil, you were telling me before that
47
was the fantasy. Yes.
The best that's possible is 45. In fact,
yes.
>> Yes. Where did we lose? Where did we we
lose the two? What's the problem here?
The problem is really that it's putting
down the fourth stick so it crosses all
the other circles in the right way. Ah,
because this black stick obviously looks
like it goes out this way towards the It
crosses It doesn't It crosses the red
stick here.
And it crosses the blue stick here. So
the
the black stick is okay, but it's also
got to cross all the circles. And the
circles have to cross all the circles.
Oh, cuz it doesn't cross the green
circle, does it? Ever. Obviously.
>> No, obviously. The black stick will
never cross the green circle. It will
cross the green stick.
>> Yeah.
Eventually, a few miles away. Not the
green circle.
>> But not the green circle. Okay. So,
we've lost two two goals. We're down by
two goals and that's the best you can
do.
Where's the fifth lollipop going to go?
We have
estimates
for that. And again, it's by taking one
of the four and making a copy of it and
perturbing it a bit.
Okay. Shaking it a bit and putting it
down. And in fact, Jonas worked out how
to do that with taking
copies of all four of these and putting
them down and jiggling them a little
bit. We have bounds with with five
circles. All we know, I mean, we know a
lot. It's either 71 or 72 regions.
But we don't know which of those two.
>> We don't know which of those two. Are
people working on this? Uh
I don't know. They might after they've
seen this video. I hope they will. We We
think the answer's probably 71.
But that's just a guess.
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>> [music]
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