Question 5

A supercell is a box in a grid that holds a number bigger than all its neighbouring numbers.

Step 1 — Understanding Supercells

Let us imagine a grid of boxes. Each box has a number inside it. A box can have other boxes next to it. These are its neighbours. A supercell is a special box. Its number must be bigger. It must be bigger than all its neighbours. If a box has no neighbours, it is a supercell.

Step 2 — Exploring Different Grids

Let us try different grid sizes. We will count the maximum supercells.

Case 1: One cell Imagine one box. It has the number 5. [ 5 ] This box has no neighbours. So, it is a supercell. Maximum supercells:

$\boxed{1}$

Case 2: Two cells Imagine two boxes in a row. Let us put numbers 5 and 3. [ 5 | 3 ] The box with 5 has 3 as its neighbour. 5 is bigger than 3. So, the box with 5 is a supercell. The box with 3 has 5 as its neighbour. 3 is not bigger than 5. So, the box with 3 is not a supercell. Maximum supercells:

$\boxed{1}$

Case 3: Three cells Imagine three boxes in a row. Let us put numbers 5, 1, 6. [ 5 | 1 | 6 ] The box with 5 has 1 as its neighbour. 5 is bigger than 1. So, the box with 5 is a supercell. The box with 1 has 5 and 6 as neighbours. 1 is not bigger than 5 or 6. So, the box with 1 is not a supercell. The box with 6 has 1 as its neighbour. 6 is bigger than 1. So, the box with 6 is a supercell. Maximum supercells:

$\boxed{2}$

Case 4: Four cells (2x2 grid) Imagine a square of four boxes. Let us put numbers like this: [ 5 | 1 ] [ 2 | 6 ] The box with 5 has 1 and 2 as neighbours. 5 is bigger than 1 and 2. So, the box with 5 is a supercell. The box with 1 has 5 and 6 as neighbours. 1 is not bigger than 5 or 6. So, the box with 1 is not a supercell. The box with 2 has 5 and 6 as neighbours. 2 is not bigger than 5 or 6. So, the box with 2 is not a supercell. The box with 6 has 1 and 2 as neighbours. 6 is bigger than 1 and 2. So, the box with 6 is a supercell. Maximum supercells:

$\boxed{2}$

Case 5: Five cells Imagine five boxes in a row. Let us put numbers 5, 1, 6, 2, 7. [ 5 | 1 | 6 | 2 | 7 ] The box with 5 is a supercell. The box with 1 is not a supercell. The box with 6 is a supercell. The box with 2 is not a supercell. The box with 7 is a supercell. Maximum supercells:

$\boxed{3}$

Case 6: Six cells (2x3 grid) Imagine a grid of two rows and three columns. Let us put numbers like this: [ 7 | 1 | 8 ] [ 2 | 9 | 3 ] The box with 7 is a supercell. (Neighbours 1, 2) The box with 1 is not a supercell. The box with 8 is a supercell. (Neighbours 1, 3, 9) The box with 2 is not a supercell. The box with 9 is a supercell. (Neighbours 1, 2, 8, 3) The box with 3 is not a supercell. Maximum supercells:

$\boxed{3}$

Step 3 — Noticing a Pattern

Let us look at our results. Number of cells | Maximum supercells ----------------|------------------- 1 | 1 2 | 1 3 | 2 4 | 2 5 | 3 6 | 3

We can see a pattern here. The number of supercells is always half the total cells. Or, it is one more than half the total cells. For example, for 4 cells, half is 2. We got 2 supercells. For 5 cells, half is 2.5. We got 3 supercells. The supercells often appear in a checkerboard pattern. Or, they are in the middle parts of the grid. Cells in the middle have more neighbours. Corner cells have fewer neighbours. Edge cells have more neighbours than corners.

Step 4 — Strategy for Maximum Supercells

We want to make many cells supercells. A supercell needs to be bigger than its neighbours. So, we should place the largest numbers. We place them in positions that have many neighbours. The middle cells have the most neighbours. So, we put the largest numbers in the middle. We make sure these numbers are bigger than their neighbours. We place smaller numbers around them. This makes the middle cells supercells. It also makes the smaller numbers not supercells. This strategy helps us get the most supercells. For example, in a 3x3 grid: [ 1 | 2 | 3 ] [ 4 | 9 | 5 ] [ 6 | 7 | 8 ] The number 9 is in the middle. It is bigger than all its neighbours. So, 9 is a supercell. This is a good way to start. We can also use a checkerboard pattern. This pattern places large numbers next to small numbers. This helps many cells become supercells.

Answer

(i) The number of supercells are more than or equal to half the number of cells. (ii) Yes, the pattern I notice is that supercells tend to occur more frequently in the middle of the grid, where a cell has more neighbouring cells to compare to. The number of maximum number of supercells can be half of the cells or one more than half (half + 1) of the total number of cells. (iii) To get the maximum number of supercells, we should place the largest numbers in the middle of the table, ensuring they are surrounded by smaller numbers. Since the middle cells have more neighbours to compare to, they are more likely to be supercells if they are larger than their adjacent cells. Smaller numbers should be placed at the edges and corners to reduce the chance of them being supercells. By strategically placing the largest numbers in central positions, we can maximise the number of supercells in the table.