[0:00] What's a lollipop? Tell me about
[0:01] lollipops. Yeah, well, this is a
[0:03] mathematical lollipop. What it consists
[0:05] of
[0:06] it is a circle and a
[0:08] on a stick.
[0:10] And if you continued the stick, it would
[0:12] pass through the center of the circle.
[0:14] This part of the stick is invisible and
[0:16] the stick is infinite. So, it's a a
[0:18] circle with a perpendicular stick coming
[0:21] out of it, perpendicular to the edge of
[0:22] the circle. That's a lollipop. And we've
[0:25] got one of them and it looks like that
[0:26] and it divides up the region. If it's If
[0:29] we're doing them in the sand, it divides
[0:31] up the beach into the part that's inside
[0:34] the circle and the rest of the beach.
[0:36] And so, one lollipop divides the beach,
[0:40] the paper, into two regions, inside the
[0:43] circle and all the rest. Because this
[0:46] goes off to infinity. What if we have
[0:48] two lollipops? This is actually a little
[0:50] tricky. If you have two lollipops, how
[0:53] many regions can you make by overlapping
[0:56] them? Well, you could do this. You could
[0:58] put your other lollipop here and its
[1:02] stick could go here. And then, if we did
[1:04] that, how many regions would we get?
[1:06] One, two, three, four, five, six. We get
[1:11] We could easily get six regions.
[1:13] >> not get seven? Does that If these go
[1:15] forever, is the other side of the thing
[1:17] not a seventh? It's seven.
[1:18] >> It is seven, right.
[1:19] >> Yeah, yeah, you're right. Yeah, there
[1:20] you go. I know my stuff. You know your
[1:22] stuff.
[1:22] >> Seven. All right. So, each stick goes
[1:24] off to infinity. So, with with two, we
[1:27] can certainly get seven. And that's
[1:29] okay, but it's not great. You can
[1:31] actually get 10 pieces if you do it
[1:33] right. And what you do is you make sure
[1:36] that the circles of the
[1:38] lollipops overlap in just a little
[1:41] sliver. And then, you make sure the
[1:43] stick of the first lollipop cuts the
[1:47] circle of the second lollipop. Okay,
[1:50] yeah. Yeah, and then, scalpel, blue,
[1:54] we get
[1:56] this. It's going to go through the the
[1:58] center imaginary through the center of
[2:01] that. It's going to cut across this gulf
[2:04] where the two circles
[2:07] and then it's going to cut that one. So,
[2:09] I'll give it a little bit of space.
[2:13] Looks like that. And what you get let's
[2:16] count the pieces. How many pieces do we
[2:18] get when we do that? It's very careful.
[2:20] We all right. We have first of all, we
[2:22] have below the the sticks and above the
[2:25] sticks. They they two infinite regions.
[2:27] All right, one, two, then three and
[2:30] four. Yeah. And then five for the
[2:32] sliver, six and seven down here.
[2:35] >> Yeah.
[2:36] And then eight, nine, 10. So, we get 10
[2:38] pieces. You can show using higher
[2:40] mathematics Euler's formula that the
[2:43] crucial thing
[2:45] if you want to get the most pieces
[2:48] what you need to focus on are the
[2:50] intersections, the crossings between one
[2:54] lollipop and the other lollipop. And the
[2:56] crossings it's it's crucial that you
[2:59] notice. You using Euler's formula you
[3:01] can show that the important thing is to
[3:04] get the most intersections between the
[3:07] lines of one lollipop and the lines of
[3:09] the other lollipop. And the
[3:11] intersections
[3:12] the there are three kinds of
[3:14] intersections. There are intersections
[3:16] where the circle part cross and you can
[3:19] see I've managed to make this circle and
[3:21] this circle cross in two points.
[3:24] There's also
[3:25] intersections where the stick of one
[3:28] lollipop crosses
[3:30] the circle of the other lollipop and
[3:33] we'd like each of them to happen twice.
[3:35] So, we can see that blue stick crosses
[3:38] the other lollipop twice. Correct. And
[3:41] vice versa, this the stick crosses this
[3:44] one here and here. And then the sticks
[3:46] themselves can cross.
[3:48] And they do here at just once. Sticks
[3:51] either cross or they don't. So, with two
[3:53] two circles, you get seven
[3:54] intersections.
[3:56] And there's a formula, the number of
[3:58] pieces equals intersections number of
[4:03] intersections plus n, the number of
[4:06] sticks plus one.
[4:07] With two, n equals two,
[4:10] we got seven intersections. I showed you
[4:13] 2 + 2 + 2 + 1, seven, and then two
[4:17] because we got two circles plus one, and
[4:19] that's 10. And that's the best we can do
[4:21] with two lollipops. And if we had three
[4:23] lollipops, ideally, we'd make each pair
[4:27] of lollipops intersect in this way. And
[4:30] it looks like this. All right. It is
[4:33] very tricky to draw. Okay, so the new
[4:35] lollipop is the green one. That's
[4:37] >> one at the bottom. Okay.
[4:39] >> Yeah. And it intersects the red one in
[4:42] the same way that the blue and the red
[4:43] intersected, and it intersects the blue
[4:45] one in the same way. Each pair of
[4:48] lollipops here
[4:50] meet in seven intersections. And the
[4:53] stick from the third new lollipop Yeah,
[4:56] it's going up. But it's slightly
[4:58] off-center, so that it doesn't it
[5:00] doesn't intersect with the intersection
[5:02] of the first two sticks. Correct.
[5:04] >> To maximize our sections.
[5:05] >> Yeah, you never want to have three
[5:06] things meeting at a point because you
[5:08] make a tiny little change and you pick
[5:11] up one piece, one region. So, that So,
[5:13] that stick is slightly off Slightly
[5:16] off-center, and it's true. You might
[5:18] think I'm fudging this, but actually, if
[5:21] you work to multiple precision and you
[5:24] draw it carefully, I did actually draw
[5:26] it carefully.
[5:27] And you can see it in the OEIS entry for
[5:30] this sequence. So, the So, how many
[5:32] pieces did we end up with for three
[5:34] lollipops? We need to know how many
[5:36] intersections there are. Each pair
[5:38] intersect in seven points. So, there are
[5:40] seven intersections there, seven there,
[5:43] and seven there. And none of them have
[5:45] been counted twice.
[5:46] >> No, they're all distinct. You can see
[5:49] check I'm very careful not to have any
[5:51] triple points or higher. So, we got 21
[5:55] intersections. All right, n equals
[5:57] three. All right. Three lollipops. The
[6:00] number of intersections is equal to 7 +
[6:02] 7 + 7 each because each we got three
[6:06] lollipops and each pair meets in seven
[6:08] points.
[6:09] >> Yeah. And they're all different. So,
[6:10] that's 21 and then the formula is this.
[6:15] We add n, which is three, and we add
[6:17] one, we get 25. So, with three
[6:20] lollipops, we get 25 regions. But, where
[6:23] are we going to put the fourth lollipop?
[6:25] >> That's all I can think of.
[6:25] >> This is
[6:26] really, really hard. Yeah.
[6:29] Cuz you cuz if you put it up You put it
[6:31] up there, it's not going to meet
[6:33] >> No. No.
[6:34] >> Yeah. No, it is really hard. Where does
[6:37] the fourth lollipop go?
[6:38] >> Where does the And I tried and I did
[6:41] various drawings. And we know what we
[6:43] want. We want the maximum number of of
[6:45] intersections between all the pairs of
[6:48] lollipops. With with four lollipops,
[6:51] we've got six intersections. So,
[6:53] ideally, we'd get 6 * 7 42
[6:57] intersections.
[6:59] Let's give people some thinking time.
[7:04] Yeah.
[7:04] >> All right. What's the answer? Well, it
[7:07] what didn't come very easily. On
[7:10] Christmas Eve, I posted a message to the
[7:12] Secret Santa mailing list explaining
[7:14] this problem and asking for help. It
[7:17] said it for I said with four lollipops,
[7:20] it's really tricky. And at 1 minute past
[7:23] midnight, I got an email from a couple
[7:26] of old friends who said that they could
[7:29] get
[7:30] 43 regions. The maximum would be 47.
[7:34] If you could get every pair to meet in
[7:37] seven points, you'd get 42 + 4 + 1,
[7:42] you'd get 47 regions. They got close,
[7:44] but
[7:45] but not not very close. And how And how
[7:49] did they uh place their circles to do
[7:50] that? Well, they took my drawing of
[7:54] three circles, and they added a fourth
[7:56] circle, which they got by perturbing one
[7:59] of the three a little bit. So, we still
[8:02] crossed most of the things the same way.
[8:04] >> They didn't perturb one of the existing
[8:06] circles. They They added
[8:08] >> They They put their new lollipop on top
[8:11] of the existing lollipop and perturbed
[8:13] that one. Yes. They took a copy of the
[8:15] red one and perturbed it a bit. Maybe
[8:17] they changed the the the diameter a
[8:19] little bit, I'm not sure. And they
[8:21] changed the angle of the stick. And that
[8:23] gave them 43. And that gave them 43
[8:26] regions.
[8:27] Cool.
[8:29] It was pretty good. Yeah. And for the
[8:31] first 12 hours, that was the world
[8:33] record. And then
[8:35] 2 minutes past noon on Christmas Day, I
[8:39] got an email from
[8:41] someone on the Sequence Fans mailing
[8:43] list who I've never met, although since
[8:45] we've talked on Zoom.
[8:49] He was able to get 44 regions. But
[8:54] later, he got it up to 45. 45 regions.
[8:57] And what's more, he proved that was
[8:59] optimal. So, there was no point in
[9:02] anyone trying to get more. You could
[9:04] theoretically have gotten 46 or 47, but
[9:08] you can't. He proved that 45 is best
[9:10] possible. Yeah. What he did was really
[9:12] extraordinary.
[9:15] He took those three, and he magnified He
[9:17] modified them a little bit. He made one
[9:20] rather bigger than the other two, about
[9:22] twice as big. And then he blew blew it
[9:25] up, magnified it by a factor of 100. So,
[9:28] these circles got really, really huge.
[9:32] And when you looked at the edge of the
[9:33] circle, it was the circle was so huge,
[9:36] it the edge looked almost like a
[9:38] straight line.
[9:39] And I will show you what those straight
[9:41] lines looked like.
[9:43] And here's a picture of how it looks
[9:46] after he's blown it up. That green line
[9:48] is part of a gigantic circle. Yeah, and
[9:51] that's the stick of the green lollipop.
[9:52] >> That's the stick of the green line.
[9:54] And the red [clears throat]
[9:55] >> And the red is also that's
[9:57] >> The red is the
[9:59] It's this
[10:00] red circle magnified so that that arc
[10:04] looks like a straight line. And there it
[10:06] is. And that's the stick. And then blue,
[10:09] this is the blue circle. Yeah. And
[10:12] here's the blue stick. So he he
[10:14] magnified it. And then he very cleverly
[10:17] put a fourth circle on top of the place
[10:20] where the sticks come together. So that
[10:22] little black lollipop, that's the new
[10:25] lollipop and it's miniaturized right in
[10:28] the mess between the other three.
[10:29] Exactly. Yes, brilliant. So that So
[10:33] there it is. And if we go back and look
[10:36] at the previous picture,
[10:38] the extra fourth lollipop, the black
[10:41] lollipop is actually here. You just
[10:43] can't see it. It's so tiny. It's in
[10:46] blots of ink where the three circles
[10:49] and the three sticks come together.
[10:52] Neil, you were telling me before that
[10:55] 47
[10:56] was the fantasy. Yes.
[10:58] The best that's possible is 45. In fact,
[11:02] yes.
[11:02] >> Yes. Where did we lose? Where did we we
[11:05] lose the two? What's the problem here?
[11:06] The problem is really that it's putting
[11:09] down the fourth stick so it crosses all
[11:12] the other circles in the right way. Ah,
[11:15] because this black stick obviously looks
[11:17] like it goes out this way towards the It
[11:19] crosses It doesn't It crosses the red
[11:22] stick here.
[11:23] And it crosses the blue stick here. So
[11:25] the
[11:26] the black stick is okay, but it's also
[11:29] got to cross all the circles. And the
[11:31] circles have to cross all the circles.
[11:33] Oh, cuz it doesn't cross the green
[11:35] circle, does it? Ever. Obviously.
[11:36] >> No, obviously. The black stick will
[11:38] never cross the green circle. It will
[11:39] cross the green stick.
[11:41] >> Yeah.
[11:42] Eventually, a few miles away. Not the
[11:45] green circle.
[11:46] >> But not the green circle. Okay. So,
[11:48] we've lost two two goals. We're down by
[11:50] two goals and that's the best you can
[11:52] do.
[12:04] Where's the fifth lollipop going to go?
[12:06] We have
[12:08] estimates
[12:09] for that. And again, it's by taking one
[12:13] of the four and making a copy of it and
[12:17] perturbing it a bit.
[12:19] Okay. Shaking it a bit and putting it
[12:21] down. And in fact, Jonas worked out how
[12:25] to do that with taking
[12:28] copies of all four of these and putting
[12:31] them down and jiggling them a little
[12:33] bit. We have bounds with with five
[12:35] circles. All we know, I mean, we know a
[12:38] lot. It's either 71 or 72 regions.
[12:42] But we don't know which of those two.
[12:44] >> We don't know which of those two. Are
[12:46] people working on this? Uh
[12:48] I don't know. They might after they've
[12:50] seen this video. I hope they will. We We
[12:53] think the answer's probably 71.
[12:55] But that's just a guess.
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