The parity of a sum or product is fixed before you compute it

0:00Here is a table with seven rows, and three of them we already know. Even plus even is even. Odd plus odd is even. Even plus odd is odd. Every one of those came from a picture rather than from examples. But the other four rows are about subtraction, and they are empty. Even take away even. Odd take away odd. Even take away odd. Odd take away even. Now, you could go and work out four more rules and memorise those too.

0:34Do not. There is nothing new down there, and I can show you why in about thirty seconds. Because subtraction cannot do anything to a leftover dot that addition has not already done. Picture a number as dots, paired off, two to a row. Now take some away. Not any old dots. Take away a whole block of complete rows. Watch what happens to the leftover at the top. Nothing. It is still sitting there.

1:05You removed pairs, and pairs are exactly the part that was never in question. So an odd number minus an even number is still odd. The stray dot survived. And an odd minus an odd? Each one has a stray. Take one away and the other goes with it. That leaves whole rows only. So the answer is even, and that is the same as the addition row. Fill all four in, and they match the addition table line for line. Subtraction brought nothing new.

1:41So here is a long sum. Six numbers, added up. What is the parity of the answer? You could add them. You do not have to. Cross out every even number in the list, because none of them can change anything. Every even number is complete rows. It brings no stray dot to the party at all. Now count what is left. Just count them. Do not add them. Four odd numbers. Four strays, which pair off, leaving nothing. So the total is even.

2:14That is the whole calculation. How many odd numbers are in the list, and is that count odd or even? And once you can do that, some questions become almost embarrassingly quick. A piggy bank. Inside it are coins worth one, coins worth five, and coins worth ten. You are not told how many of each. You are told something much thinner. There is an odd number of the one-coins. An odd number of the five-coins.

2:46And an even number of the ten-coins. Somebody claims the whole lot comes to two hundred and five. Can that be right? The one-coins: an odd count of them, each worth one. That is an odd amount. The five-coins: odd times odd is odd, so that is an odd amount too. And the ten-coins are even, whatever their count. Odd, odd, even. Two strays, which pair up. The total is even. And two hundred and five is not.

3:20Now something that looks like a different subject, and is not. Rule a rectangle into small squares. Three rows by three columns. Count them. Nine. Which is odd. Now add a column. Three by four. Twelve small squares. Even. One extra column, and the answer changed parity. So the count is a product, and we are asking about the parity of a product. And that is a question we have never actually answered.

3:52But the grid is about to answer it for us, and rather more convincingly than the times table would. Take the three by four grid and try to cover it in dominoes. Two squares each. Because four is even, every row splits into exactly two dominoes. Nothing is left over. It did not matter that the other side was three. It would work if it were three hundred. One even side, and the whole grid falls into pairs.

4:24Now the three by three. Try the same thing. You get four dominoes down, and one square that will not pair with anything. It is the stranded dot again, wearing a different hat. Two odd sides leaves exactly one over. Any even side leaves none. That is the entire rule. Which means we can now do something that ought to feel like cheating. Twenty-seven by thirteen. Is the number of small squares odd or even?

4:57Both sides odd. So it is odd. I have not multiplied anything. Forty-two by seventy-eight. Forty-two is even, so the grid is even. Done. A hundred and thirty-five by six hundred and fifty-four. Six hundred and fifty-four is even. So it is even. That is the whole working. For the record, that last grid has eighty-eight thousand two hundred and ninety small squares. But we knew its parity without going anywhere near that number, and that is the point.

5:33Now the interesting bit, where this stops being tidy. Here is an expression. Three n plus four. Feed it a number and it gives you back a number. What is its parity? You might say even, because four is even. Let us just try it. When n is three, it gives thirteen. Odd. When n is eight, it gives twenty-eight. Even. When n is ten, thirty-four. Even again. So it has no answer. Sometimes odd, sometimes even, and which one depends on what you feed it.

6:10Compare that with a hundred p. Whatever p is, a hundred p is even. Or forty-eight w minus two. Always even. Every value, no exceptions. So what is different? Look at the number in front of the letter. A hundred is even. Forty-eight is even. And three is not. An even number times anything is even, so an even coefficient wipes the letter out completely. Whatever n is doing, it cannot get through. The constant on the end decides everything on its own.

6:47But three is odd. Odd times n keeps n's parity, so the letter is back in charge. Even coefficient, the parity is fixed. Odd coefficient, the parity follows the letter, and it moves. One more thing about these expressions, and it is the one people slide past. Take six k plus two. Six is even, so this one has a fixed parity. Always even. Its values start at eight, then fourteen, then twenty.

7:20So here are the even numbers, and let us light up the ones this expression reaches. Eight. Fourteen. Twenty. Twenty-six. Look at everything still dark. Two, four, six, ten, twelve, sixteen, eighteen. Ten of the first fourteen even numbers never come out of it. It only lands on every third one. Always even is a statement about what it gives you. Reaching all the evens is a completely different claim.

7:53So which expression does reach every even number? Write them out. Two, four, six, eight, ten. The first is two. The second is four. The third is six. Every one of them is twice its own position. So the nth even number is two n. Now write the odd numbers underneath, lined up position by position. One, three, five, seven, nine. And look at the gap between the rows. One, all the way along. Every odd number sits exactly one below the even number above it.

8:29So the nth odd number is two n minus one. The hundredth even is two hundred, and the hundredth odd is a hundred and ninety-nine. Last thing. Here are four claims, and exactly one of them is true. Four m minus one always gives odd numbers. Four m is even, take one, always odd. True. Every even number can be written as six j minus four. We have just seen that trick. False.

9:01Two f plus three gives both odd and even values. Two f is even, plus three. Always odd. False. And the fourth: two p plus one and two q minus one both list all the odd numbers. Two q minus one starts at one, then three, five. That reaches all of them. But two p plus one starts at three. It never produces one, so it misses the very first odd number.

9:32And notice what settled each of those. One value, chosen to break the promise. Always means always.