Why testing values one by one is not a method, and what replaces it

0:00Two hundred coins in a purse, and packets that cost thirty coins each. Buy x packets and you spend thirty x, and that has to stay under two hundred. Before touching any of it, notice what x is allowed to be. It counts packets, so it is a whole number, and it is not negative. That decision is made by the situation, and it is made before the algebra starts.

0:28It matters more than it looks: the same statement, with x allowed to be something else, has a different answer. The obvious way in is to try numbers. Substitute one, see whether the statement comes out true, write down the verdict. That works, and it is where the word solution comes from. But it settles numbers one at a time, and the thing we actually want is a whole collection. That gap is the subject.

1:00So start where anyone would start. Substitute nought, then one, then two, and keep going. The left-hand side comes out nought, thirty, sixty, ninety, one hundred and twenty, one hundred and fifty, one hundred and eighty. Every one of those is under two hundred. Seven substitutions, seven verdicts of true. Then seven packets. Thirty sevens is two hundred and ten, which is over. The eighth verdict is false. And here almost everyone stops, and writes down: nought to six.

1:36Which is the right answer. But what entitles you to stop there? Eight substitutions give eight facts about eight numbers. Nothing in that column says a word about eight, or nine, or four hundred. It is safe because of something about this expression, and it is worth saying out loud. Watch the two sides as the count goes up. The left one climbs by thirty every step; the right one does not move at all.

2:08So the gap between them grows, at every single step, all the way out to forty and past it. Once the left side has gone over, it cannot come back under. That is what licenses the stop. Put it to the test: look above the failure, out as far as forty, and ask how many solutions are sitting there. None. Stopping at seven throws away nothing at all, and the largest count that works is six.

2:37Notice that this is now a stronger claim than before. Not the largest one I found; the largest one there is. Two things had to hold together: the sides must keep pulling apart, and the search above must come back empty. Neither alone is worth anything. At three the gap is already growing, but the search finds solutions above it, so the answer is no; over a stretch where the search comes back empty but the sides pull the wrong way, no again.

3:13That is a fact about this expression, not about tables. Here is what happens when you forget the difference. Take an expression that climbs and then comes back down, and ask when it is over twenty. Substitute nought upward and the values run nought, eleven, twenty, twenty-seven, thirty-two, thirty-five, thirty-six, then thirty-five, thirty-two, twenty-seven, twenty, eleven, nought. It rises to a peak and falls away. The turn happens at exactly one count.

3:48Now run the habit. Substitute nought: the value is nought, which is not over twenty. False. The very first row fails. Stop there, as the habit says, and you have just thrown away seven real solutions: three, four, five, six, seven, eight and nine. Every one makes the statement true, and the method never looked at any of them. Ask this expression for its largest solution and the answer is a refusal, not a number. Nothing here says it stays false, because it does not.

4:25So the stopping rule is not a rule of tables. It is a licence, and it has to be earned by the expression each time. Back to the packets, and a distinction that carries the rest of this. Four is a solution. Substitute it and the statement comes out true. Seven is not a solution. The solution set is different: it is not a number at all, it is the whole collection of numbers that work.

4:56Six checks certify six numbers, not a collection - and the collection is what was asked for. And there is a second thing hiding in the word. Ask whether a half is a solution here. The question refuses to be answered, and it should. Half a packet is not something x was ever allowed to be. Ask whether nought is a solution, when counting starts at one, and it refuses for the same reason.

5:25So a solution set is never a fact about a statement alone. It is a fact about the statement and what the letter was allowed to be, together. Which means one statement can have more than one answer. Keep thirty x under two hundred and change only what x may be. Let x be a counting number starting at one, and six values work: one through six. Let counting start at nought instead, and seven work.

5:56Now let x be any whole number, negatives included. Search from minus twelve up to forty and nineteen turn up. But nineteen is a fact about where the search stopped, not about the statement. Push the bottom of the search down and the number goes up, forever. Ask for a solution below nought and you get minus one. Ask for one below minus a thousand and you get minus a thousand and one.

6:26There is no bound you can name that the answer does not reach past. Three collections, three answers, one statement. And the largest solution is six in all three of them. So what is different between them? The ends. An answer can have a smallest member, a largest, both, or neither - four separate questions. Counting from one: the answer has both. Its smallest is one and its largest is six.

6:58And notice where that smallest came from. It is not the statement's doing. Counting simply starts there. Counting from nought: both again, with nought at the bottom. All the whole numbers: a largest, and no smallest at all. Ask for one and the answer is a refusal. And it goes the other way too. Take a statement built the other way round, and over the whole numbers it has a smallest, minus one, and no largest.

7:29Each end is settled by its own argument. Having no last member tells you nothing whatever about the first. Now take the guard rails off completely and let the letter be any number on the line. Something breaks at once. The line has no step from one member to the next, and no smallest member. So ask for the answer as a list and you get a refusal - not a long list, not a truncated one.

7:59Ask which ends the answer has, by stepping along it, and that refuses too, because there is nothing to step by. Take a statement whose answer is everything under two. Start at nought, which is a solution. Halfway between nought and two is one, which is a solution and is bigger. Do it again: halfway between one and two is one and a half. Again: one and three quarters. Every one of those is a solution, and every one is bigger than the last, and the construction never runs out.

8:35So there is no largest solution to write at the end of a list. There is no end of the list. It is worse than having no end, and this part finishes testing off for good. Take two solutions, as close together as you like. One and three quarters, and one point nine nine. Halfway between them is one point eight seven, and that is a solution too. It sits strictly between the two.

9:05You could not have written a list that goes from one to the other, because there is always another one in the gap. No first entry. No last entry. And no next entry anywhere in the middle either. No amount of patience turns that into a table. So substituting has to be replaced, and not with more substituting. What replaces it is rewriting: turn the statement into a different statement whose answer you can read off.

9:35Which raises the only question that matters about a rewrite. How do you know it did not change the answer? Do not assume. Measure. Take a statement with the letter on both sides: five x minus three, under three x plus one. Now hand in six moves and score them all the same way: apply the move, compare the two statements at every whole number from minus six to six, and count the disagreements.

10:08Add three to both sides: no disagreements. Take three x from both sides: none. Halve both sides: none. Multiply both sides by minus one and turn the symbol round: none. Four of the six keep the answer exactly where it was. The other two do not. And notice what happened: nobody declared which moves are legal. Each was tried and scored, and the scoring separated them - which is the difference between a rule you were handed and one you can check.

10:45Look at the two that failed, because they fail in instructive ways. Multiply both sides by minus one and leave the symbol alone. That disagrees with the original at twelve of the thirteen numbers tried. It is not slightly wrong; it is wrong almost everywhere. But it agrees at one of them, and the one is two. So a student who checked x equals two and nothing else would have certified a move that is wrong twelve times over.

11:17One test cannot certify a move, for exactly the reason one test cannot certify an answer. The other failure is quieter. Multiply both sides by nought, and every statement becomes nought against nought. It disagrees at eight of the thirteen. And take minus twelve x, over thirty. Divide both sides by minus twelve and leave the symbol alone. That one disagrees at every single number tried. Seventeen out of seventeen. Turn the symbol round as well and it disagrees at none.

11:54Each of the four symbols has exactly one that reads it from the other side, and swapping to it is what makes the negative move safe. So put three safe moves together and see what they buy. Five x minus three, under three x plus one. Add three to both sides. Take three x from both sides. Halve both sides. What comes out is x on its own, under two. Nothing left to work out; the answer is written on the page.

12:29And that chain agrees with the statement it came from at every whole number from minus twenty to twenty. Not a single disagreement. Over the whole numbers the largest solution is one, and there is no smallest. Over the line there is no largest either, by the halving argument from before. Same statement, two collections, two answers, and the rewrite did not care which. That is the whole point of a rewrite that keeps the answer.

13:01One more, to show it is not a fluke. Twenty-four x under a hundred, over the counting numbers, gives exactly four: one, two, three and four. Finally, the case that finishes the argument, because testing cannot reach it at all. Minus twelve x, greater than thirty, with x a counting number. Substitute one: minus twelve is not greater than thirty. False. Two is worse, three worse again. A tester keeps going, keeps failing, and never learns why: no failure tells you to stop, because every failure looks like the last.

13:44The answer is empty. Not small. Not hard to find. Empty. And proving that takes the same two things as before. Counting has a smallest member; the statement is already false there; and from there the two sides pull further apart every step. So it can never come back, and there is nothing to find. Now change one thing: let x be any whole number. Suddenly there is an answer, with a largest member at minus three.

14:16And ask whether that one is empty and the question refuses, because the whole numbers have no smallest member to start the argument from. Same statement. One collection gives nothing at all, the other gives an answer running downward without end. Substituting told you what a solution is, and it can never be taken away. What it cannot do is finish.