00:01
Okay, so we'd like to figure out how we can fill in these slots to make the sentence true, namely so that each of these occurrences is true based on how we fill in the rest of the sentence.
00:14
So let's start by making a few observations that are going to be able to get us started.
00:19
Well, first off, well, we can note that each of these has to be at least one, because, well, each number already appears once in at least one.
00:29
Well, it already appears once in the sentence just by how the sentence is given.
00:32
So each of these has to be at least one.
00:34
Well, that tells us to start off that none of them can be zero.
00:38
So we can't fill in a zero anywhere else.
00:41
We've already got one zero.
00:43
So this tells us that this has to be one, because we can't fill in a zero anywhere else.
00:48
Okay, so that gets us started.
00:50
And then let's make some other observations.
00:54
Well, once we fill in the sentence, fill in these total of tensile.
00:59
Slots, well, there should be 20 numbers in the sentence.
01:03
Should be 20 numbers in sentence.
01:11
Well, what does that tell us? so the slots here, the sum of these slots is going to be the number of, the number of numbers in the sentence because it's the number, if you add up all these, it's going to add up the occurrences of each of the numbers.
01:31
So that tells us that slots should add to 20.
01:41
Okay, so that's going to be useful.
01:44
Okay, so now we'd like to figure out how we can fill in the rest of this.
01:51
So first off, let's start at the bottom.
01:54
So let's see.
01:54
So if there were, so we want to see, well, can we put a nine anywhere? else in the sentence.
02:02
Is there going to be a nine anywhere else or does this have to be one? well, it seems like, well, since this has to be, um, the slots have to add up to 20, there certainly isn't going to be more than one nine in here anywhere.
02:16
So this can be at most two.
02:17
So let's suppose that it's two and let's try to show that it actually has to be one.
02:21
So let's suppose that it's two.
02:23
Um, well, uh, we've already got, um, um, this tells us that, well, we've already got two twos.
02:33
Oh, if we put another two here, then that, then we've already got three two.
02:36
So if we have two twos, that actually tells us that we have to have three twos, at least three twos.
02:44
So, so we can fill in that as like sort of a guess.
02:49
It's like a hypothetical based on this, this assumption.
02:53
Well, okay.
02:54
And then if we had two, if we had two nines, well, the sure isn't going to be more than one nine.
03:00
There aren't going to be nine of any other numbers.
03:05
So there can have most, if there was a nine anywhere else, it would have to do that there are nine ones.
03:12
But let's just see already where we're at.
03:16
So if we had these hypothetical values, then just adding them up, where one plus nine is 10, plus three is 13, plus two is 15, and we've got to fill in these other slots with something that's at least one.
03:32
So, but we can't do that.
03:34
So we already have 15.
03:35
So then this would need, you know, adding up the rest of these values, this would need to be at least 16, 17, 18, 19, 20, 21.
03:41
But this needs to be, this can be at most 20.
03:42
So we can't, so we can't do that.
03:45
So what have we shown? we've shown that this bottom slot has to be one.
03:51
There can only be one nine.
03:56
So let's try to, well, let's try to keep going.
04:00
Well, you can do a very similar thing with the eights and show that there can be a most one eight.
04:08
So, and then continuing on, well, let's try to figure out what we can do with the rest.
04:16
So it seemed like we're going to have a fair few ones.
04:21
We've already got three, and we know that we can only add up to 20, and we're going to have...