Odd and even as "can this be arranged in pairs"

0:00Five empty boxes, with plus signs between them, and thirty after the equals sign. And here is a pile of number cards to fill them with. Thirteen. Nine. One. Seven. Eleven. Five. Three. There are plenty of each, so you can use a number as often as you like. Put one card in every box, and make the five of them add up to thirty. That is the whole puzzle, and it looks like the sort of thing you solve by trying.

0:31So try it. Stop the video here if you want a proper go. I will wait, and then I am going to tell you something slightly annoying. How did you get on? Twenty-nine, probably. Or thirty-one. You can get very close. You cannot seem to land on it. Now look at the pile again, before you pick anything up. Thirteen, nine, one, seven, eleven, five, three. Every single one of those is odd. There is not an even card in the pile.

1:04That is not decoration. That is the entire puzzle, sitting in plain sight. But we cannot use it yet, because right now odd is just a word we sort numbers with. So let us make it into something we can actually look at. First though, how bad is the searching option? Seven values, five boxes, repeats allowed. That is four hundred and sixty-two different ways. You could sit down and grind through every last one of them. I did.

1:37None of them makes thirty. Not one of the four hundred and sixty-two. And here is the thing that should bother you about that answer. It took four hundred and sixty-two checks, and at the end of it I still cannot tell you why. If somebody hands me an eighth card tomorrow, I have to start again from nothing. A reason would not care about the eighth card. So let us go and find one.

2:06Take a number. Any number at all. Draw it as that many dots on the board. Now pair them off. Two dots to a row, and keep going until you run out. Six goes down as three neat rows, and nothing is left in your hand. Ten goes down as five rows. Nothing left over. Eighteen goes down as nine rows. Again, nothing. That is what even is. Not a label. A shape.

2:35It is a number that lies down flat in twos, with none left standing. And once you can see it, you can start doing things with it. Now seven. Pair them off the same way. One, two, three rows. And one dot left in your hand, with nobody to go with. I am going to draw that dot in a different colour, because it is the important one. Eleven does the same thing. Five full rows, and one dot stranded.

3:06Nineteen. Nine full rows, and one dot stranded. So odd is not a different kind of shape at all. It is exactly the even picture, with exactly one dot over. Never two. Never none. One stranded dot. That is the whole of what odd means, from here on. Right. So what actually happens when you add two odd numbers together? Draw them both. Two blocks of paired rows, and each one has a stray dot on top.

3:39Now slide the two pictures together, so they become one picture. Watch the two stray dots. They are the only loose things in the room. And there are two of them. So they pair up with each other. That new row is the argument. One green dot and one purple dot, sitting together. Every dot is now in a pair. Nothing is stranded. The total is even. Two odd numbers add up to an even number. Always.

4:11Now, I want to stop and be careful about what just happened. Because that was not an example of anything. That was a proof. Look back at what we drew and ask which numbers we used. We never said how many rows were in either block. We never needed to. The blocks could be three rows or three million. The picture is the same picture. All it says is: full rows, plus one stray. Full rows, plus one stray.

4:41And the two strays find each other, whatever is underneath them. A picture that only worked for the numbers drawn would be an illustration. This one never mentions them. So let us push our luck. What about three odd numbers? Three blocks. Three stray dots. Ignore the rows underneath, they are all paired already. Two of the strays find each other. And one is left standing. So three odd numbers add up to something odd.

5:13Four odd numbers. Four strays. Two pairs. Nothing left standing, so the total is even. Five strays gives two pairs and one stranded. Odd. Six gives three pairs. Even. Do you see what we are actually doing? We are pairing off the strays. So the question is only ever: how many odd numbers are there? The numbers themselves never come into it. Which is everything we need. So let us go back to those five empty boxes.

5:47Five cards. Every card odd. So every card contributes one stray dot. Five strays. Pair them off: one pair, two pairs, and one dot left standing. One dot with no partner means the total is odd. Thirty is even. So thirty is not reachable, and never was. And notice how much that argument just covered. It did not check four hundred and sixty-two combinations. It did not check any. It settled all of them at once, including every one nobody has ever tried.

6:25Here is exactly the same move again, wearing completely different clothes. Martin and Maria are brother and sister, born exactly one year apart, on the same day. Somebody tells you their two ages add up to a hundred and twelve. Could that be right? Try it. Fifty-one and fifty-two gives a hundred and three. Too small, and a bit odd looking. Now think about what one year apart actually forces.

6:55The counting numbers go even, odd, even, odd, all the way up. They alternate. So of any two neighbours, exactly one is even and exactly one is odd. One even, one odd. One stray dot between them. Their ages always add to an odd number. A hundred and twelve is even. So no two ages one year apart can make it. Fifty-five and fifty-six give a hundred and eleven. Fifty-six and fifty-seven give a hundred and thirteen.

7:31The claim falls straight down the gap between them. And this is where the shared birthday earns its place in the story. It is there so that both ages are whole numbers. Without it you could say fifty-five and a half and fifty-six and a half, and you would be right. Now, we have used this idea five or six times and never named it. Whether a number is even or odd is called its parity. That is all the word means.

8:02So parity is a good tool. But I want to be honest about what kind of tool it is. It rules things out. That is what it does, and it does it completely. Wrong parity means impossible. Not unlikely, not hard to find. Impossible. But turn the whole thing around, and it collapses immediately. Suppose a total does have the right parity. What has that told you? Nothing at all. Watch. Could five of our cards add up to three?

8:35Three is odd, so parity has no objection whatsoever. And it is still impossible, because five cards cannot total less than five. Parity closes doors. It never opens one. And knowing which of those you have just done is most of the skill.