Always, sometimes, or never: one counterexample settles it

0:00Here is a claim. Add two even numbers and you get a multiple of 4. Is that true? There are only three answers, and they are not three levels of confidence. Always true. Sometimes true. Never true. They are three different shapes of claim, and each one is earned by a completely different kind of evidence. Getting that straight is the whole job, and it comes before any arithmetic at all.

0:29So let us take the three of them one at a time. Start with always, because it promises the most. Always true means it holds for every case the claim speaks about. Every single one. And there is no end to them. Whatever collection of numbers you have checked, there are more you have not. So an always claim cannot be established by examples. Not by ten, not by a thousand.

0:58It has to be argued. You need a reason that covers cases you will never write down. But watch what happens to it in the other direction. Because one single failure is enough to destroy it completely, and that asymmetry is the most useful thing in this whole topic. Let me show you how badly examples can mislead you. Take a number, multiply it by itself, add the number again, then add 41.

1:29Start at zero and you get 41, which has no factors except 1 and itself. Then 43. Then 47. Then 53. Keep going. Every single answer has no factors except 1 and itself. It holds for the first case, and the second, and the tenth, and the thirtieth. Forty cases in a row, and not one failure. At that point you would be forgiven for calling it always true. Forty greens is a lot of green.

2:02And then you try the next one. The forty-first case gives 1681. And 1681 is 41 times 41. So it does have a factor. The claim is false. Not weakened. Not usually right. False, as stated. One case did that, and one case is all it ever takes, because always means every and a single exception is enough to break every. Notice that the forty successes were not wrong. They were just never evidence in the first place.

2:42Which is why a counterexample is such a powerful thing to look for. It is the only cheap move in the game. Now sometimes true, and this is the verdict people misuse most. Sometimes true does not mean I am not sure. It is a definite answer with a definite price. To earn it you must produce two things. One case where the claim works, and one case where it fails.

3:11Here is a claim. If a number is divisible by 8, then 8 divides any two numbers that add up to it. 72 is divisible by 8. Split it as 48 and 24. Eight goes into both of those. The claim worked. Now split the same 72 as 50 and 22. Eight goes into neither of them. The claim failed. Two splits of the same number, and between them they settle it. There are 71 ways to split 72, and only 8 of them have both parts divisible by 8.

3:52Here is another one, and it needs the same treatment. If a number is divisible by 7, is it divisible by every multiple of 7? Take 42, which is 7 times 6. Is it divisible by 14? Fourteen is 7 times 2, and yes, 42 is three fourteens. Is it divisible by 28? Twenty-eight is 7 times 4, and no, it is not. One that works, one that does not, off the same number. Sometimes true, and now it is earned rather than guessed.

4:30Of the first twelve multiples of 7, only four of them go into 42. Now two claims that look like twins, and get different verdicts. First: a number divisible by 9 and by 4 must be divisible by 36. Second: a number divisible by 6 and by 4 must be divisible by 24. Same shape. Two divisors, multiply them, claim the product. Below 601 there are 16 numbers divisible by both 9 and 4, and every single one of them is a multiple of 36. That one is always true.

5:13There are 50 numbers divisible by both 6 and 4, and 25 of them are not multiples of 24. The very smallest of them, 12, already breaks it. So the shape of a claim tells you nothing about its verdict. You have to actually settle each one. That leaves never true, and never is just as sweeping as always. It says the thing cannot happen. Not once, ever. So it needs a positive argument. Not searching and failing to find one is not an argument at all.

5:51Here is the claim: an odd number added to an even number gives a multiple of 6. The short reason first. Every multiple of 6 is even. An odd number added to an even number is odd. An odd number cannot be an even number, so the two can never meet. That is a clash, and a clash is what a never verdict is made of. Across 1600 pairs of one even and one odd, it does not happen once - but the sweep is not why we believe it. The clash is.

6:28Let us write that clash out properly, because there is a trap waiting in it. Write the even number as 2n and the odd number as 2m plus 1. Set the total equal to a multiple of 6, which is 6j. Move the 1 across. Two n plus two m equals six j minus 1. The left is 2 times something, so it is even. The right is a multiple of 6 with 1 taken off, so it is odd.

6:59Even equals odd. There is no such pair, and now we know why. Now the trap. Why not write them as 2n and 2n plus 1, and save a letter? Because that forces them to be next-door neighbours. Out of the 1600 honest pairs, only 40 are neighbours - so one letter would have settled a much smaller question than the one being asked. Claims often arrive with somebody's name attached, and a name is not evidence.

7:33First claim. Take any three numbers that each leave 2 when divided by 6, and add them. Always a multiple of 6. That one holds, and here is why. The three leftovers are 2 and 2 and 2, which make exactly 6. Second claim. A number leaving 8 on division by 12, added to one that is 4 short of a multiple of 12, gives a multiple of 8. Twenty and eight make 28, which is not a multiple of 8. But 8 and 8 make 16, which is. Sometimes true.

8:10Third claim. Reverse the digits of a multiple of 9 and it stays a multiple of 9. 1089 becomes 9801, and that one is always true, because reversing does not change what the digits add up to. Fourth claim. Some multiples of 11 stay multiples of 11 when you double them, and some do not. That one is false, and in an unusual way: all 300 in the sweep do. There is no some do not to find.

8:47So here is what a properly written verdict actually looks like. For always: the argument. Not examples, a reason. For sometimes: two numbers. One that works and one that does not, and you have to show both. For never: the clash. Two quantities forced to be equal that cannot be, like an even one and an odd one. And if you find a counterexample, do not stop there. Ask what condition would rescue the claim.

9:18Divisible by 6 and by 4 does not give 24. But it does give 12, every time. That is the productive answer. The original claim was wrong, and there was a true one hiding just behind it. One last thing, and it is the most grown-up idea here. Sometimes the honest answer is that you have not decided yet. That is not the same as sometimes true. Sometimes true is a finished verdict with two witnesses standing behind it.

9:53Not decided means you are still looking, and saying so is exactly right. What is not right is calling something always true because you got tired of checking. Remember the forty cases in a row. That claim looked settled for a very long time, and it was never settled at all. Three verdicts, three kinds of evidence. Knowing which one you owe is most of the work.