Supercells

Look at any row or grid of numbers. Some cells stand out because they are the biggest in their tiny neighbourhood. Those cells have a special name: supercells.

Concept

A supercell in a row of numbers is a cell whose number is bigger than the number in every cell directly touching it. In a one-dimensional row, "touching" means the cell to the immediate left and the cell to the immediate right. So a cell is a supercell if it is strictly bigger than both its neighbours (or its only neighbour if it sits at the end).

For example, in the row:

$3, 8, 5, 2, 9, 4, 7$

• The first cell ($3$) has only one neighbour ($8$). Since $3 < 8$, this is not a supercell.
• $8$ has neighbours $3$ and $5$. $8 > 3$ and $8 > 5$. Supercell!
• $5$ has neighbours $8$ and $2$. $5 < 8$. Not a supercell.
• $2$ has neighbours $5$ and $9$. Not a supercell.
• $9$ has neighbours $2$ and $4$. Supercell!
• $4$: neighbour $9$ is bigger. Not a supercell.
• $7$ at the end: only neighbour is $4$. $7 > 4$. Supercell!

So the supercells in this row are $8$, $9$, and $7$.

Some observations:

• Two supercells cannot stand next to each other. If two neighbours were both supercells, each would have to be bigger than the other , impossible.
• A supercell can appear at the end of a row, because there is only one neighbour to beat.
• The biggest number in the row is always a supercell (it beats both its neighbours).

In a two-dimensional grid (a table of numbers), a supercell is one that is bigger than all its neighbours , left, right, above, and below (and sometimes diagonals, depending on the rule we set).

Designing a row or grid with a chosen number of supercells is a puzzle. With $7$ cells you can have at most $4$ supercells (every other cell). With $n$ cells you can have at most $\lceil n/2 \rceil$ supercells.

Worked examples

Example 1. Identify supercells in the row $4, 1, 6, 3, 2, 5$.

• $4$: only neighbour $1$. $4 > 1$. Supercell.
• $1$: neighbours $4, 6$. Not a supercell.
• $6$: neighbours $1, 3$. $6 > 1$ and $6 > 3$. Supercell.
• $3$: neighbours $6, 2$. Not.
• $2$: neighbours $3, 5$. Not.
• $5$: neighbour $2$. $5 > 2$. Supercell.
• Supercells: $4, 6, 5$.

Example 2. Place the numbers $1, 2, 3, 4, 5, 6, 7$ in a row so that there are exactly $3$ supercells.

• We want supercells at positions $1, 3, 5$ (or any non-adjacent set of $3$).
• Try: $7, 1, 6, 2, 5, 3, 4$.
• Check: $7 > 1$ supercell; $1 < 7, 6$ not; $6 > 1, 2$ supercell; $2 < 6, 5$ not; $5 > 2, 3$ supercell; $3 < 5, 4$ not; $4 > 3$ supercell.
• Oops, that gives $4$ supercells. Adjust to get exactly $3$.
• Try: $5, 1, 6, 2, 7, 3, 4$. $5 > 1$ ✓; $1 < 5, 6$ ✗; $6 > 1, 2$ ✓; $2 < 6, 7$ ✗; $7 > 2, 3$ ✓; $3 < 7, 4$ ✗; $4 > 3$ ✓. That is $4$ supercells.
• For exactly $3$, end with a smaller number: $5, 1, 6, 2, 7, 4, 3$. Check the last: $3 < 4$, so $3$ is not a supercell, and $4 < 7$, also not. Supercells: $5, 6, 7$. Done , exactly $3$.

Example 3. What is the maximum number of supercells in a row of $10$ cells?

• Supercells cannot be adjacent, so at most every other cell.
• $10$ cells $\Rightarrow$ at most $5$ supercells.

Example 4. In a $3 \times 3$ grid (with only horizontal/vertical neighbours), is the centre cell ever a supercell? When?

• The centre has $4$ neighbours: up, down, left, right. To be a supercell, it must be bigger than all $4$.
• This is possible if the centre is the biggest number in the grid (or biggest among those $5$ cells).

Try it yourself

1. List supercells in: $2, 9, 3, 5, 8, 1, 7, 4$.
2. Build a row of $6$ cells using $1$–$6$ with exactly $2$ supercells.
3. Build a row of $7$ cells using $1$–$7$ with the maximum number of supercells.
4. Can a row have $0$ supercells? Hint: think about the largest number.
5. In the row $1, 2, 3, 4, 5$ (already sorted), find all supercells.
6. In a $2 \times 2$ grid using $1$–$4$, how many supercells can there be?
7. Challenge: In a row of $100$ random numbers, roughly what fraction would you expect to be supercells?

Activity

Supercell puzzle. On graph paper, draw a $5 \times 5$ grid. Fill it with the numbers $1$ to $25$ in any order you like , each number used once. Now circle every supercell (bigger than each of its four side-neighbours). How many did you get? Can you fill the grid in a way that gives more supercells? Fewer?