Testing a given list, and what "finite" versus "infinite" changes

0:00Here is a list of numbers. One, one, one, two, two, two, three, three, three. The only question on the table is whether it is an arithmetic progression. Last time we said what that means: one fixed number, added again and again, all the way along. So the work is to find out whether such a number exists for this list. There is a way to find out, but a strange one: it can only ever give you a definite no.

0:30A definite yes is not on offer, and you will see exactly why not. The test is short. Take each entry away from the one that follows it, all the way along, and look at what you get. Four, ten, sixteen, twenty-two. Ten minus four is six. Sixteen minus ten is six. Twenty-two minus sixteen is six. Three sixes, and nothing has gone wrong. Later minus earlier, every time, because the order carries the direction.

1:05One, minus one, minus three, minus five. Minus one minus one is minus two. Minus three minus minus one is minus two. Minus five minus minus three is minus two again. That is the whole procedure. Subtract along the list, and compare what comes out. Now watch what the two possible outcomes are worth, because they are not worth the same. Suppose two neighbouring differences disagree. Then no single number gets you from each entry to the next, because two different numbers were needed at two different places.

1:47That is not evidence. That is a proof, and nothing later in the list can rescue it. Now suppose they all agree. What have you got? Agreement on the entries you were shown, and on nothing else. There is a next entry you have not been shown, and it can be anything at all. So the test is lopsided. Failing it is conclusive. Passing it is a report on how far you looked.

2:16Which is why the list we opened with is worth doing slowly. One, one, one, two, two, two, three, three, three. First difference: one minus one is nothing. Second difference: one minus one is nothing again. Two agreements. If you were going to stop, this is where you would stop. Two differences, both nothing. Third difference: two minus one is one. And that is the end of it. Nothing, nothing, one. It is not an arithmetic progression.

2:50The trap is not that the list is difficult. It is that the list is patient. A list can agree with you a hundred times and break on the hundred and first. Compare that with a list that gives itself away at once. Minus two, two, minus two, two, minus two. First difference: two minus minus two is four. Second difference: minus two minus two is minus four. Four and minus four. The very first comparison you can make already fails.

3:24This list does have a pattern, and a strong one. It alternates, and you could carry on writing it for ever without thinking. But a pattern is not the test. The differences swing between two values instead of holding one, so it is not an arithmetic progression. Being predictable and being arithmetic are different properties. There is a shortcut you will meet. It is genuinely useful, and it is the easiest thing here to misuse.

3:57The shortcut says: if you already know a list is an arithmetic progression, one pair of neighbours is enough to tell you the step. That is true, and obviously so: if every step is the same, any one of them is the step. The misuse is to run it backwards and treat one agreeing pair as a reason to believe the list is arithmetic in the first place. I swept three thousand two hundred and eleven lists and asked, for each one, whether its first difference matches its last.

4:32Two thousand one hundred and ninety-seven of them said yes. Only a hundred and sixty-nine of those were arithmetic progressions. The other two thousand and twenty-eight were not. So the shortcut will hand you a number for a list that has no step at all. It answers what, once you have settled whether. Now the dots. A list with three dots on the end is making a claim, and it is worth being precise about which claim.

5:04The dots do not stand for entries. They stand for a rule, and the rule is the thing you were not shown. Take four entries that pass the test, and ask what the fifth could be. A hundred and sixty-nine arithmetic lists of four entries, each followed by every one of eighty-one candidate fifth entries. Thirteen thousand six hundred and eighty-nine futures, and in every one the visible four still pass the test.

5:35Thirteen thousand five hundred and twenty of those futures are not arithmetic progressions. Exactly one candidate in each eighty-one keeps the list going, and it is the one that takes the step once more. But nothing you can see tells you it is the one that was meant. When a list ends in dots and we agree to trust the pattern, that is a convention between us, not a deduction.

6:04Some lists have no dots at all. They stop, and where they stop is written down. Heights, one hundred and forty-seven up to one hundred and fifty-seven, one at a time. Eleven entries, and then it is over. Temperatures rising a tenth at a time, from minus three point one to minus two point five. Seven readings. A balance falling fifty at a time from nine hundred and fifty down to fifty. Nineteen entries.

6:36Twelve prizes, two hundred up to seven hundred and fifty, fifty apart. Ten months of saving, fifty up to five hundred. Every one of those is an arithmetic progression, and every one of them stops, which means each owns something an endless list cannot: a last entry. That sounds like bookkeeping. It is not. Having a last entry is what later lets you total a progression up without ever working out its step.

7:09And the other kind runs on. One, two, three, four, dots. A hundred, seventy, forty, ten, dots. Minus three, minus two, minus one, nought, dots. Three, three, three, three, dots, which never moves at all and is an arithmetic progression with a step of nothing. Minus one, minus one point five, minus two, minus two point five, dots. All five of those are arithmetic progressions and none of them has a last entry.

7:42Infinite here does not mean enormous. It means there is no last one. You cannot ask for the final entry of the second list. The answer is not that it is very small; there simply is not one. Same rule, same test. What differs is whether the list has an end, and that decides what you can do with it afterwards. Here is where appearance goes wrong, in both directions. Root two, root eight, root eighteen, root thirty-two.

8:16You cannot subtract those as they stand. Simplify first. Root eight is two root two. Root eighteen is three root two. Root thirty-two is four root two. Written that way the list is one, two, three, four lots of root two, and the differences are all root two. It is an arithmetic progression. Now the list that looks tidier. Root three, root six, root nine, root twelve. Root nine is three. Root twelve is two root three. Root six will not simplify at all.

8:52So the entries are root three, root six, three, two root three, and those differences do not agree. The first list looks like a jumble and is a progression; the second looks like a system and is not. You cannot compare until you have simplified, and you cannot decide until you have compared. The same trick, played with squares. One squared, three squared, five squared, seven squared. That is one, nine, twenty-five, forty-nine.

9:25Differences: eight, sixteen, twenty-four. They grow. Not an arithmetic progression. The odd numbers are a progression. Their squares are not; squaring is the operation that destroys it. Now: one squared, five squared, seven squared, and then seventy-three. That last entry is not written as a square, and the temptation is to assume it should have been, and expect eighty-one. Subtract instead. One, twenty-five, forty-nine, seventy-three. Differences: twenty-four, twenty-four, twenty-four. It is an arithmetic progression, with a step of twenty-four, and the one that broke the visual pattern is the one that made it work.

10:13Appearance is not the test. Subtraction is the test. Sometimes you are handed a situation instead of a list, and you have to make the list yourself. A fare that charges fifteen for the first kilometre and eight for every one after. Fifteen, twenty-three, thirty-one, thirty-nine. Step of eight. Arithmetic. Digging a well, where the first metre costs a hundred and fifty and each metre after costs fifty more. A hundred and fifty, two hundred, two hundred and fifty, three hundred. Step of fifty. Arithmetic.

10:55Now a pump that takes away a quarter of whatever air is still in the vessel at each stroke. Start full. After one stroke, three quarters left. After two, nine sixteenths. After three, twenty-seven sixty-fourths. The amount removed shrinks every time, because a quarter of less is less. And money earning interest on its own interest does the same thing the other way up. Ten thousand growing by eight per cent a year gains eight hundred in the first year, eight hundred and sixty-four in the second, nine hundred and thirty-three point one two in the third.

11:37A fixed percentage is not a fixed amount. Both are lists with a rule, and neither is a progression. None of that was taken on trust. A hundred and sixty-nine arithmetic lists were built, of eight entries each, using addition and nothing else. Then each was damaged six more times, once at each position from the third entry to the last, so that where a list breaks was known before anything looked at it.

12:10Three thousand two hundred and eleven lists, and three separate tests over all of them: reading the differences; a test with no subtraction in it, checking each entry against the two around it; and rebuilding the list from its first entry and one step. All three agreed, everywhere. Then the measurement that matters. For every list, and every depth from three entries up to eight, what the first few entries say was recorded beside what the whole list says. Nineteen thousand two hundred and sixty-six of those pairs.

12:49Seven thousand six hundred and five times, the entries you can see say yes and the whole list says no. That is the entire subject of this video. And not once did the visible entries say no while the whole list said yes. That is the asymmetry, measured rather than asserted. The test that stops after two agreeing differences went through the same line, and got two thousand five hundred and thirty-five wrong.

13:21So: you are handed a list, and you subtract along it. If two neighbouring differences disagree, you are finished, and the answer is no. If they all agree, you have confirmed the entries in front of you and guessed about the rest. One pair tells you the step once you know there is one, and never that there is one. Simplify before you compare, and do not trust how a list looks.

13:47And if the list stops, it has a last entry, which will turn out to matter more than it sounds.