What makes a number a perfect cube, and the three-identical-groups test

0:00Before it was a number, a cube was a shape. A solid box with every edge the same length. Here is the smallest one we will use. One unit along each edge. Call it a unit cube. Now the question that turns the shape into arithmetic. If I build a bigger cube out of these, how many do I need? Start with an edge of two. A lot of people say six, and six is the wrong six. Six is how many faces a cube has, and that never changes.

0:35Build it instead. A floor of four. Two by two. Then a second floor of four on top. Two floors of four. Eight. Eight unit cubes, and now the number two cubed means something you can hold. Once you see it as floors, the rest is easy. Edge three. Each floor is three by three, so nine, and there are three floors. Twenty-seven. Edge four. Each floor is sixteen, and four of them. Sixty-four.

1:08Edge five gives one hundred and twenty-five. And notice what that is. It is exactly how you counted a square, with one more direction to go in. A square is so many rows of so many. A cube is so many floors of a square. The shape is doing the multiplying for you. Now put them on a line and look at where they are not. One. Eight. Twenty-seven. That is every cube up to thirty.

1:39There is nothing at all from ten to twenty-six. Seventeen numbers in a row, and not one of them is a cube. Squares are thin on the ground. Cubes are far thinner. Under a hundred there are ten squares and four cubes. Under a thousand, thirty-one squares and ten cubes. And the gaps say why. Between the squares they go three, five, seven, nine, climbing by two each time. Between the cubes they go seven, nineteen, thirty-seven, sixty-one.

2:13Those are not creeping upward. They are running. The way we write it. Three cubed. A small three, up on the shoulder. It means three used three times as a factor. Three times three times three. Twenty-seven. Two things it does not mean, and both of them catch people. It is not three times three. Two threes give nine, and we wanted three of them. And it is not three times the number. Cubing is not tripling.

2:46The little number counts how many times the big one is used. That is all it ever is. It works on anything you can multiply. Four sixths, cubed. Cube the top, cube the bottom. Sixty-four over two hundred and sixteen. A decimal. Thirteen point nought eight, cubed. Do it in whole numbers and put the point back afterwards. Two decimal places, cubed, give six, so the point goes six from the right. Two thousand two hundred and thirty-seven point eight one nought one one two.

3:24And a negative. Minus six, cubed. Minus six times minus six is plus thirty-six. Times minus six again is minus two hundred and sixteen. That is worth stopping on. Squaring throws the sign away. Cubing keeps it. A negative number has a cube, and its cube is negative. Now a warning, because something you learned about squares does not carry over. A square can only end in six of the ten digits. Nought, one, four, five, six, nine.

4:01So four digits are enough to throw a number out on sight. Ends in seven? Not a square. Done. Try the same thing on cubes. One ends in one. Eight ends in eight. Twenty-seven ends in seven. Sixty-four ends in four. A hundred and twenty-five ends in five. Keep going and every single digit turns up. All ten of them. So there is no digit you can throw a number out on.

4:30That test, which cost nothing and settled things instantly, is worth exactly nothing here. But look closer, because something better is hiding in it. Cube the ten digits and write down what each one ends in. Nought gives nought. One gives one. Two gives eight. Three gives seven. Four gives four. Five gives five. Six gives six. Seven gives three. Eight gives two. Nine gives nine. Every answer is different.

5:04So the last digit of a cube tells you the last digit of the number it came from. One to one, no ties. Squares cannot do that. Two and eight both square to something ending in four, and three and seven both end in nine. The digit that was useless for rejecting cubes turns out to be the one that identifies them. One more thing the shape decides. Take a number ending in a zero. It has a factor of ten in it.

5:36Cube it, and that ten gets used three times too. Ten cubed is a thousand. So a factor of ten in the number becomes a factor of a thousand in the cube. Which means the zeros at the end of a cube arrive three at a time. None, or three, or six, or nine. Never one. Never two. So if somebody hands you a number ending in exactly two zeros and asks whether it is a cube, you can answer without looking at the rest of it.

6:08Squares do this too, but two at a time, which is a different answer to a different question. Now the test, and if you have met the one for squares you already know it. Take three thousand three hundred and seventy-five. Break it into primes. Three, three, three, five, five, five. Deal them out. But this time into three piles, not two. Three, three, five. Three, three, five. And three, three, five.

6:38All three the same. Multiply one of them out. Fifteen. And fifteen used three times over is three thousand three hundred and seventy-five. You can read the same six primes the other way, as two triplets. Three cubed times five cubed. Either reading gives you the edge without ever going looking for it. So why three piles? Because if a number is something used three times over, then every prime that something owns has been handed over three times. Once by each copy.

7:17So every prime turns up a number of times that three divides. And that is exactly what lets you split them three ways. It runs back the other way too. Three matching piles, multiplied together, give the number. And underneath sits the same thing as before. A number has only one set of primes. Break it apart in any order you like and the same primes come back, so there is no second reading in which the triplets could come out differently.

7:50Which leaves one sentence for the whole of this. Two piles tests for a square. Three piles tests for a cube. The pile count is the little number on the shoulder. Nothing else changed. And when it fails, it tells you why. Five hundred. Two, two, five, five, five. Deal them into three. Two and five. Two and five. And a lonely five. Not the same, so five hundred is not a cube.

8:20And look at the reason. The five turns up three times, which is fine. The two turns up twice. Twice is one short of a triplet. So we know what it wants. One more two. Five hundred times two is a thousand, and a thousand is ten cubed. Try another. One thousand three hundred and twenty-three is three, three, three, seven, seven. The threes are a triplet already. The sevens are one short.

8:51Multiply by seven and you get nine thousand two hundred and sixty-one, which is twenty-one cubed. Notice it is not always the small prime that is missing, and not always the big one. It is whichever one three does not divide. Finish with a number that belongs to both stories. Sixty-four. Eight eights are sixty-four, so it is a square. Four fours are sixteen, times four again is sixty-four, so it is a cube.

9:24Coincidence? Break it up. Sixty-four is two, six times over. And six splits evenly into two piles of three, and evenly into three piles of two. That is the whole reason. The count of each prime has to be divisible by two for a square, and by three for a cube. So to be both, it has to be divisible by six. The next one is seven hundred and twenty-nine. Three, six times over. Twenty-seven squared, and nine cubed.

9:58They are the sixth powers, and there is nothing lucky about any of them. Two piles, three piles. Count the primes, and the shape tells you the rest.