Rearranging a sum so it can be done in your head

0:00Here is a figure from your book. It looks like a puzzle, but it is really a permission slip. Down the middle, five numbers. Twenty-five thousand. Four hundred. Thirteen thousand. Fifteen hundred. Sixty thousand. Down the left and the right, eight targets. And arrows running from the middle out to each one. Your job is to hit each target by adding numbers from the middle column. And you may use any of them as often as you like.

0:31That last rule is the whole game. The middle column is not five numbers. It is five numbers, repeatable, which is a far bigger toolbox than it looks. Your book works two of them for you. Start with thirty-eight thousand eight hundred. Twenty-five thousand. Plus four hundred. Plus four hundred again. Plus thirteen thousand. Twenty-five and thirteen thousand is thirty-eight thousand. Two four hundreds is eight hundred. Thirty-eight thousand eight hundred.

1:07And notice what did not happen. Nobody wrote out a column and carried anything. The second one is three thousand four hundred. Fifteen hundred, plus fifteen hundred, plus four hundred. Three thousand, and four hundred. Done. Which is the point of this whole section, and it is worth saying plainly. Mental arithmetic is not the same sum done faster. It is a shorter sum with the same answer. Now try twenty-eight thousand yourself. And then keep going, because there is more than one road.

1:46Twenty-five thousand, plus fifteen hundred, plus fifteen hundred. Three parts. Or thirteen thousand, plus thirteen thousand, plus five four hundreds. Seven parts. Same destination. More than twice the work. Sixty-three thousand tells the same story. Sixty thousand plus two fifteen hundreds is three parts. Or twenty-five thousand, plus twenty-five thousand, plus thirteen thousand. Also three. So when your book asks you to find a way, the honest follow-up is: is that the cheapest way? Usually it is not.

2:28Then your book asks a much better question. Can you make one thousand? Have a go. You will not manage it, and here is the reason. Every part except four hundred is already bigger than a thousand. So the only tool you have is four hundred. Four hundred. Eight hundred. Twelve hundred. It steps straight over one thousand and never lands on it. So one thousand is out of reach. And then the book asks which whole thousands are impossible.

3:01The answer is lovely. Just that one. Two thousand you can make: five four hundreds. Three thousand you can make: two fifteen hundreds. And once you can make two and three, you can make everything above them. Four is two and two. Five is two and three. Six is three and three. And so on, for ever. So out of every whole thousand there is, exactly one is unreachable. One thousand.

3:32Second half of the section, and your book hands you a new toolbox with a new rule. Six numbers this time. Forty thousand. Seven thousand. Three hundred. Fifteen hundred. Twelve thousand. Eight hundred. And now you are allowed to subtract as well as add. That sounds like a small change. It is not. Because until now every move took you closer from below, and you had to land exactly. Now you can go past the target and come back. Which means the numbers near your target become useful even when they are too big.

4:12Here is the book's example, and it is the best argument for the new rule. The target is thirty-nine thousand eight hundred. By adding only, you would have to build that out of twelve thousands and seven thousands and fifteen hundreds. Several goes. With subtraction, you start at forty thousand. Take away eight hundred. Thirty-nine thousand two hundred. Add three hundred. Add three hundred again. Thirty-nine thousand eight hundred. Four moves, and the first one carried you almost the whole way.

4:48Try forty-five thousand the same way. Forty thousand, plus twelve thousand, minus seven thousand. Three moves. Overshoot and correct. Once you have seen it, you use it everywhere. Next, the book changes the question. Instead of hitting a target, it asks about the shape of the answer. It prints two sums. Twelve thousand three hundred and fifty, plus twenty-four thousand five hundred and forty-five, is thirty-six thousand eight hundred and ninety-five.

5:24Five digits plus five digits gave five digits. And forty-eight thousand nine hundred and fifty-two, minus twenty-four thousand five hundred and forty-seven, is twenty-four thousand four hundred and five. Again five. Then it hands you a grid. Each row names how many digits the two inputs have, and how many the answer has, and asks for an example. And some of those rows have no example at all. Those are the interesting ones.

5:58Because to show a row is impossible, an example is no use to you. You need an argument about the largest and the smallest. Take the sharpest one. Can two four-digit numbers add up to a six-digit number? The biggest four-digit number is nine thousand nine hundred and ninety-nine. So the biggest sum you could possibly get is that, plus itself. Nineteen thousand nine hundred and ninety-eight. Five digits. Not six. Not ever.

6:31You have just ruled out every pair of four-digit numbers there is, in one line, without testing a single one. That is the difference between here is an example and here is a reason, and it is the habit the exercise is training. Now the next section, which is the same idea drawn instead of written. Your book puts six pictures in front of you and asks for the total of each.

7:00Every one of them could be counted item by item. That is not what they are for. Each picture is arranged, and the arrangement is a hint about how to group. Look at the shape first. Decide what the repeated part is. Then the counting almost disappears. Let me do three of them with you. First picture. Five rows of numbers. Row one, four forties. Row two, five fifties. Row three, four forties. Row four, five fifties. Row five, four forties.

7:38You could add twenty-two numbers one at a time. Or you could notice there are only two kinds of row here. Three rows of four forties. That is twelve forties. Four hundred and eighty. Two rows of five fifties. Ten fifties. Five hundred. Four hundred and eighty plus five hundred. Nine hundred and eighty. Two multiplications and one addition, instead of twenty-one additions. Second picture, and this one is engineered.

8:11On top, a block of four rows of eight thirty-twos. Underneath, four rows of four sixty-fours. Count the top block. Four times eight is thirty-two, so there are thirty-two thirty-twos. Thirty-two thirty-twos is one thousand and twenty-four. Now the bottom. Four rows of four is sixteen, so sixteen sixty-fours. Sixteen sixty-fours is also one thousand and twenty-four. The two blocks weigh exactly the same. Half as many pieces, each worth twice as much.

8:49So the total is two thousand and forty-eight, and you barely did any arithmetic. You noticed a trade. Last, the pictures made of dots and rings, and a warning that comes with them. One is an eight-by-eight grid of dice faces. Twenty of them show five dots. The other forty-four show one. Twenty fives is a hundred. Forty-four ones is forty-four. A hundred and forty-four. And then there is a bullseye. One thousand at the centre. Around it a ring of four five hundreds. Around that, eight two fifties.

9:29And you can feel a pattern arriving. The value halves as you go out, the count doubles, so every ring after the centre is worth the same. Two thousand. Which would make the outer ring sixteen one-twenty-fives. Two thousand again, and seven thousand altogether. Except go and count that outer ring on the page. There are eighteen discs in it, not sixteen. So the outer ring is two thousand two hundred and fifty, and the real total is seven thousand two hundred and fifty.

10:05The pattern was nearly true. And nearly true is wrong. Which is the last thing this section has to teach. Group by the shape, yes. But count the shape first. Next time, the same regrouping habit, used to estimate rather than to work things out exactly.