Supercells: a number's status comes from its neighbours

0:00Here is a row of eight numbers, and three of the cells have been shaded in. Forty-three. Seventy-nine. Seventy-five. Sixty-three. Ten. Twenty-nine. Twenty-eight. Thirty-four. And the shaded ones are seventy-nine, twenty-nine, and thirty-four. Your book calls them supercells and leaves you to work out what makes one. Pause and stare at it, because it is a better puzzle than it looks. Here is what trips everybody. Look at thirty-four. It is shaded, and it is the second smallest number in the row.

0:38And seventy-five is not shaded, even though it is more than twice as big. So whatever a supercell is, it is not simply being big. Here is the rule. A cell is a supercell if its number is bigger than the numbers in the cells right beside it. That is all of it. Beat your neighbours. Take seventy-nine. On its left, forty-three. On its right, seventy-five. Seventy-nine beats both, so it is shaded.

1:10Now seventy-five. On its left is seventy-nine, which is bigger. That is enough. Seventy-five is out. It does not matter that seventy-five is big. It lost to the cell next door. Which is exactly the move we made last time, with the line of children. A supercell is not a size. It is a position in a landscape. Now, what about the cells at the two ends of the row?

1:39A cell in the middle has two neighbours. A cell at the end has one. Plenty of people assume that makes the ends harder. It is the opposite. An end cell has one rival to beat instead of two. It is the easiest place in the row to be a supercell. Look at thirty-four again, at the right-hand end. Its only neighbour is twenty-eight. Thirty-four beats twenty-eight. Done. That is how a number as small as thirty-four gets in, while seventy-five does not.

2:14Fewer rivals, easier win. Let's test the rule on every cell. Forty-three, at the left end. One neighbour, seventy-nine, and it is bigger. Not a supercell. Seventy-nine. Beats forty-three, beats seventy-five. Supercell. Seventy-five loses to seventy-nine. Out. Sixty-three loses to seventy-five. Out. Ten loses on both sides. Out. Twenty-nine. Its neighbours are ten and twenty-eight. Beats both. Supercell. Twenty-eight loses to twenty-nine. Out. Thirty-four, at the end, beats twenty-eight. Supercell.

2:58Three supercells: seventy-nine, twenty-nine, thirty-four. Exactly the three shaded cells. Your book gives a second row, and there are two traps in it. Two hundred. Five seventy-seven. Six twenty-six. Three forty-five. Seven ninety. Six ninety-four. A hundred and nine. A hundred and ninety-eight. First trap: two hundred, at the left end. An end cell, one rival — but that rival is five seventy-seven, and it wins. So two hundred is out.

3:35Being at an end makes it easier. It does not make it automatic. Second trap: six ninety-four. A big number, and it loses to seven ninety on its left. Out. While a hundred and ninety-eight, at the far right, beats a hundred and nine, so it is in. The shaded cells are six twenty-six, seven ninety, and a hundred and ninety-eight. Put a hundred and ninety-eight beside six ninety-four. The small one is a supercell. The much bigger one is not.

4:11This is where it stops being a marking exercise and becomes mathematics. Can two supercells sit side by side? Suppose they could. Call them A and B, right next to each other. A is a supercell, so A beats everything beside it — including B. So A is bigger than B. But B is a supercell too, so B beats everything beside it — including A. So B is bigger than A.

4:39Both cannot be true. A number cannot beat a number that beats it. So two supercells can never be neighbours. Not in this row, not in any row, not ever. And notice we proved that without looking at a single actual number. That one impossibility does a lot of work. Your book asks how many supercells a row of nine cells can have. No two of them can be next to each other. So if cell one is a supercell, cell two cannot be.

5:12The best you could possibly do is take every other cell. One, three, five, seven, nine. That is five. And there is no sixth, because it would have to sit next to one of those five. So five is a hard ceiling for nine cells. For a row of any length, it is the length divided by two, rounded up. A real theorem, from one line of reasoning about A and B.

5:41A ceiling is only half an answer. Can you actually reach five? Yes, and the recipe is simple. Go high, low, high, low, all the way along, starting high. Nine thousand, then one hundred, then nine thousand and one, then a hundred and one. Every high cell has a low cell on each side, so every high cell wins. Cells one, three, five, seven and nine. So five is both the most you can have and something you can build. That is what finishing a problem looks like.

6:17Compare that with sorting the row from smallest to largest. Then every cell loses to the one on its right, except the last. Same nine numbers. One supercell instead of five. The arrangement is doing all the work. Two more results fall straight out of this. Take whatever the biggest number in the row is. Wherever it sits, its neighbours are smaller — they have to be, it is the biggest.

6:47So the largest number always wins its own cell. Every row of different numbers has at least one supercell. Now the smallest number. Whatever sits beside it is bigger, so it always loses. The smallest can never be a supercell. Try it on the first row. The biggest is seventy-nine, and it is shaded. The smallest is ten, and it is not. Two guarantees, without knowing the arrangement. Now your book moves into a grid. The rule does not change — what changes is who counts as beside you.

7:27In a grid, your neighbours are left, right, above and below. And here is something students get backwards, thinking of a chessboard. In a grid, a corner cell has two neighbours. An edge cell has three. A cell in the middle has four. So corners are the easiest place to be a supercell, and the middle is the hardest. Your book gives a four-by-four table and shades four cells in it.

7:54Apply the rule yourself and five cells qualify. The bottom-left one, five thousand seven hundred and eighty-five, beats both of its neighbours too. The book never claims the shading is a complete list. So trust the rule, not the colouring. Then comes the best exercise in the section. It runs backwards. You get a grid with most of its cells blank. Every number in it must be five digits long, using one, zero, six, three and nine exactly once each.

8:31And you are told which cells are shaded. Fill the blanks so that those cells, and only those, come out as supercells. Watch what that forces. The top-left cell is shaded, and sitting next to it is ninety-six thousand, three hundred and one. So the top-left has to beat it, using those same five digits and no others. There is exactly one number that manages it: ninety-six thousand, three hundred and ten.

9:02One cell, one possible answer — from the shading, not from arithmetic. The rest of the grid has room to move. That corner does not. Last one — the puzzle your book closes the chapter with. A three-by-three grid of five-digit numbers, with exactly one supercell in it: sixty-two thousand, eight hundred and seventy-one, in the middle. Which makes sense — the middle cell has four neighbours, and it beats all four.

9:34Now the task. Swap two digits inside one of the numbers, and leave the grid with four supercells. This one is worth doing properly, because there is exactly one swap in the whole grid that works. Take that middle number and swap its two and its one. It becomes twelve thousand, eight hundred and seventy-six. The middle cell collapses. And it had been blocking all four cells around it. So the four cells that touch it — above, below, left and right — every one of them becomes a supercell.

10:11One swap. Four comparisons change. The count goes from one to four. And that is the fragility the section is really teaching. A supercell is not a fact about a number. It is a fact about a neighbourhood. Next time, we put a very large number on a number line without counting — just by feel.