A constant difference is the signature of a linear pattern

0:00Here is a pattern of square tiles, growing one stage at a time. Stage one is a single square. Stage two: a row of two, with one square resting on top of the right-hand one. Three tiles. Stage three: two rows of two, and the same single square on top. Five tiles. Stage four: three rows of two, and the cap. Seven tiles. One, three, five, seven. The question is not what comes next. Almost anyone can say nine. The question is how many tiles stage fifty needs, without drawing forty-six more pictures.

0:44Put the counts in a row and keep going. One, three, five, seven, nine, eleven, thirteen. Now look at what happens between them, rather than at them. The counts themselves are all different, so there is nothing to notice there. The steps might not be. One to three is two. Three to five is two. Five to seven is two. Every gap is two, and the gap never changes. That is the whole signal. A quantity whose gaps are all the same size is called linear, and a constant gap is the fingerprint you look for.

1:22But knowing the gap is not the same as knowing the rule. Watch how easy it is to get that step wrong. It goes up by two each time. So the rule is two n. That is the most natural guess in the topic, and it is wrong. Try it. Two times one is two. Two times two is four. Two times three is six. Two, four, six. Compare with one, three, five.

1:50The gaps are right. Every gap is two, exactly as it should be. The values are wrong, every single one of them. And they are wrong by the same amount every time: one too many. The gap tells you what multiplies the stage number. It tells you nothing at all about where the pattern starts. So stop reading the numbers and look at the picture again. Every stage is the same two shapes. A block that is two tiles wide, and one lone square capping the right-hand column.

2:27At stage one the block has no rows at all. Only the cap. At stage two the block has one row. At stage three, two rows. At stage four, three rows. So at stage n the block has n minus one rows. And each row holds two tiles. Which gives the count directly. Two times n minus one, for the block, plus one for the cap. Check it at stage four. Two times three is six, plus one is seven. Seven tiles, which is what we counted.

3:03Now open the bracket. Two n minus two, plus one, is two n minus one. There is the rule, and every piece of it came off the drawing. The two is the width of the block. The n minus one counts the rows added since stage one. The minus one that survives is the price of starting late. Compare that with the guess. Two n was right about the growth and silent about the start. This one knows both.

3:36A rule earns its keep by answering questions the drawing cannot. Stage fifteen. Two times fifteen is thirty, minus one, twenty-nine tiles. Stage twenty-six. Fifty-two minus one. Fifty-one tiles. Now run it the other way. Somebody hands you twenty-one tiles and asks which stage that is. Two n minus one is twenty-one. So two n is twenty-two, and n is eleven. Forty-seven tiles the same way. Forty-eight, halved, is stage twenty-four.

4:14And you can always check a backwards answer forwards. Two times twenty-four is forty-eight, minus one, forty-seven. It holds. Now try forty-eight tiles. Two n minus one is forty-eight. Two n is forty-nine. And n is twenty-four and a half. There is no stage twenty-four and a half. So no stage has forty-eight tiles. That is not a failure of the arithmetic. It is the rule telling you something true.

4:48Two n is even, and one less than an even number is odd. Every count this pattern can produce is odd. So half of all the numbers you could name are unreachable, and the rule knows which half. Nothing so far required the pattern to grow. Someone starts with a hundred put aside and spends five of it every day. Day zero, a hundred. Day one, ninety-five. Day two, ninety. Day three, eighty-five. Day four, eighty.

5:21The gaps are minus five, minus five, minus five, minus five. Constant. Negative, but constant, and that is all the definition asks for. A hundred to start, minus five for each day. Day twelve leaves forty. Day fifteen leaves twenty-five. And on day twenty there is nothing left. Notice this record begins at day zero, so the starting amount is sitting right there in the data. The tile pattern had no stage zero. Where the counting starts is a fact about the situation, not about linearity.

6:02Now a case that does not behave. A ride opens at twenty-five, and that covers the first two kilometres. After that, each further kilometre adds fifteen. So the first two kilometres cost the same as each other, because they are both covered by the opening charge. One kilometre, twenty-five. Two kilometres, still twenty-five. Three kilometres, forty. Four, fifty-five. Five, seventy. Six, eighty-five. Look at that table for a moment before we do anything to it.

6:40Take the gaps. Twenty-five to twenty-five is zero. Then fifteen. Then fifteen, fifteen, fifteen. Zero, fifteen, fifteen, fifteen, fifteen. Those are not all the same. This is not a constant-difference run. But it very nearly is. Cover the first entry, and everything left is perfectly regular. The pattern is linear from the second kilometre onward, and not before. Which means the honest description is not a rule by itself. It is a rule together with the stretch it covers.

7:20Build the rule on the part that behaves. Twenty-five to begin, plus fifteen for every kilometre past the second. Twenty-five plus fifteen times n minus two. Ten kilometres. Ten minus two is eight, and eight is the number of increments you pay for. Not ten. Fifteen eights are a hundred and twenty. Plus twenty-five is a hundred and forty-five. Open the bracket and it tidies to fifteen n minus five.

7:52Now feed that tidy rule one kilometre. Fifteen minus five is ten. The actual fare is twenty-five. So the restriction is not caution and it is not decoration. Outside its range the rule does not go quiet. It gives a confident answer that is simply false. One last thing, because the gap being two hid it. Take a rectangle whose length is pinned at thirteen, and let the breadth run twelve, ten, eight.

8:26The areas are a hundred and fifty-six, a hundred and thirty, a hundred and four. The gap is minus twenty-six. But the breadths did not step by one. They stepped by minus two. And thirteen times minus two is minus twenty-six. The gap in the output is the multiplier times the step you chose in the input. Once more, going up. A box with a seven by eleven base, and heights five, nine, thirteen.

8:58The volumes are three hundred and eighty-five, six hundred and ninety-three, a thousand and one. The heights step by four. The base area is seventy-seven. And seventy-seven fours are three hundred and eight, which is exactly the gap. So a constant gap is a fingerprint, and the size of that gap is the multiplier scaled by your own step size. Find the gap, then ask what the start costs you, and then ask where it stops being true. All three, every time.