Turning a pattern into a formula that predicts

0:00Here is a machine, and the interesting thing about it is the part you cannot see. Two numbers go in at the top. One number comes out at the bottom. In between is a sealed box, doing the same thing every single time. Five and two go in, and eight comes out. Eight and one go in, and fifteen comes out. Nine and eleven give seven. Ten and ten give ten.

0:27Now, before writing a single symbol, say what it is doing in ordinary words. It doubles the first number, and then takes the second one away. That sentence is the whole rule, and it was found by looking rather than by algebra. Only now name them. Call the first one a and the second one b, and the machine is two a minus b. That is a candidate, so run it against every case you were given.

0:58Two fives are ten, take away two, eight. Correct. Two eights are sixteen, take away one, fifteen. Correct. Two nines are eighteen, take away eleven, seven. Correct. Two tens are twenty, take away ten, ten. Correct. And the fifth machine, the one with nothing in its box. Six and four. Two sixes are twelve, take away four. Eight. Notice something the words already warned you about. The order of the two inputs matters.

1:31Nine and eleven gives seven, but eleven and nine gives thirteen. Only the first slot gets doubled. Here is a second machine, and it is the more important one. Four and one give five. Six and nought give one. Three and two give seven. The fourth box, ten and three, was left empty. Find the rule. Multiply the two numbers, and add one. Check it. Five, one, seven. It fits.

2:02So the empty box is thirty one. Except, hold on. Try this instead. Thirteen, take away twice the first number. Thirteen take eight is five. Thirteen take twelve is one. Thirteen take six is seven. That fits too. And that second rule never once looks at the second input, which those three cases had no way of telling you. One says thirty one and the other says minus seven. Fitting what you were shown is how you find a candidate, never how you finish.

2:39Patterns that repeat are gentler, and they are worth doing slowly. A border runs along a wall: the same few tiles, over and over. Number the positions from one, left to right. Now look at what is actually there. Position four is the same design as position one. Position five matches position two. Position six matches position three. So there are not six designs on this wall. There are three, and then they start again.

3:12Call them design A, design B and design C, in the order they first turn up. The border is A, B, C, A, B, C, and it keeps going. Everything worth asking about it now comes out of a single number. Three. Where does design C land? First at position three. Then six. Then nine, then twelve. Those are the multiples of three, and there is a formula hiding in them.

3:42The nth time design C appears, it is at position three n. Read that slowly, because it is the commonest place to slip. The letter n is not the answer. It counts which appearance you are asking about. The expression is what hands you back the position. So n equals seven means the seventh C, and the formula says it sits at position twenty one. Which is worth having, because nobody drew twenty one tiles.

4:15Design B is easier than it looks, because it never has to be built from scratch. B turns up at two, five, eight, eleven, fourteen. Every one of those is one less than a multiple of three. So the nth B sits at three n minus one. And A, which comes one before B, sits at three n minus two. Three formulas, each a single shift from the next, all standing on the same three.

4:45Now here is that slip in the flesh. Put n equal to forty one into three n minus one. That gives one hundred and twenty two. One hundred and twenty two is the position of B's forty first appearance. It is not the hundred and twenty second appearance of anything. Now run the whole thing backwards. Somebody hands you a position and asks which design sits there. Position ninety nine. Divide it by three.

5:19Thirty three, remainder nought. A remainder of nought means you landed exactly on a multiple of three, and that is design C. Position one hundred and twenty two. Divide by three: forty, remainder two. Remainder two means the second design in its group. That is B. Position one hundred and forty eight gives forty nine, remainder one. Design A. So remainder one is the first design and remainder two is the second.

5:51And nought is the third, not the first, which is the step that catches almost everybody. Now a pattern that grows instead of repeating. Matchsticks laid out as a strip of triangles, each one sharing a side with the next. Step one is a single triangle. Three sticks. Step two is two triangles, and here is the thing to be careful about. Two separate triangles would need six sticks. This needs five, because one side is shared.

6:26Step three needs seven, not nine. The counts run three, five, seven, nine, eleven, thirteen. Now somebody asks for step one hundred and eight. You could draw it. One hundred and eight triangles in a row, and then count them stick by stick. You are not going to do that, and that reluctance is exactly where formulas come from. So build one out of how the pattern actually grows. Step one is three. Every step after that adds two more.

7:00Step two is three plus two. Step three is three plus two plus two. Step four is three, and then three twos. So at step y, the number of twos is one less than y itself. Three, plus two times, bracket, y minus one. Now the wrong guess, because almost everybody makes it once. It grows by two, so the formula is two y. Put y equal to one. That says two sticks, and the picture plainly has three.

7:32Getting the growth right is only half the job. The starting point has to be right as well. Here is a second way of looking at that same strip, and it gives a different formula. Sort the sticks by direction. Some lie flat, and the rest lean. At step two there are two flat ones and three leaning ones. At step three, three flat and four leaning. At step four, four and five.

8:01So at step y there are y flat sticks, and y plus one leaning ones. Altogether, two y plus one. But the first way gave three plus two times, bracket, y minus one. Two formulas, and only one strip. So do not go hunting for a step where they disagree. Simplify. Three plus two y minus two is two y plus one. The same formula all along, because the three of step one was always one plus two.

8:35Now spend it, because this is the thing the formula was for. Step thirty three. Two thirty threes are sixty six, plus one. Sixty seven sticks. Step eighty four. One hundred and sixty nine. Step one hundred and eight. Two hundred and seventeen. Nobody drew any of those, and nobody needed to. And that is the real test of a formula. Not whether it agrees with the cases you were shown.

9:06It has to agree with those, obviously, or it is simply wrong. But agreeing with them is where it starts. A formula earns its keep on the cases nobody checked. Say it in words, name the quantity that moves, write it down, and then ask it something you do not know.