Question 24

We will first find the current supercell and then test different digit swaps to create four supercells.

Step 1 — Find the current supercell

Let us look at the numbers in the grid. A number is a supercell if it is bigger than all its 8 neighbours.

Original grid: 16,200 | 39,344 | 29,765 23,609 | 62,871 | 45,306 19,381 | 50,319 | 38,408

Let us check each number:

• 16,200 (top-left): Neighbours are 39,344, 23,609, 62,871. It is smaller than all of them. So, it is not a supercell.
• 39,344 (top-middle): Neighbours include 62,871. It is smaller than 62,871. So, it is not a supercell.
• 29,765 (top-right): Neighbours include 39,344, 45,306, 62,871. It is smaller than all of them. So, it is not a supercell.
• 23,609 (middle-left): Neighbours include 39,344, 62,871, 50,319. It is smaller than all of them. So, it is not a supercell.
• 62,871 (center): Its neighbours are 16,200, 39,344, 29,765, 23,609, 45,306, 19,381, 50,319, 38,408. Let us compare 62,871 with all its neighbours: 62,871 > 16,200 (True) 62,871 > 39,344 (True) 62,871 > 29,765 (True) 62,871 > 23,609 (True) 62,871 > 45,306 (True) 62,871 > 19,381 (True) 62,871 > 50,319 (True) 62,871 > 38,408 (True) So, 62,871 is the only supercell. This matches the problem statement.

Step 2 — Find the swap to create 4 supercells

If a number is a supercell, it must be greater than all its 8 neighbours. This means two supercells cannot be next to each other (they cannot be adjacent horizontally, vertically, or diagonally). If they were, each would have to be greater than the other, which is impossible.

In a 3x3 grid, the only way to have 4 cells that are not next to each other is if they are the four corner cells. Let us call the corner cells:

• Top-Left (TL): 16,200
• Top-Right (TR): 29,765
• Bottom-Left (BL): 19,381
• Bottom-Right (BR): 38,408

For these four corner cells to become supercells, each of them must be greater than its 3 neighbours. For example, TL's neighbours are 39,344, 23,609, and 62,871. So, TL must be greater than 62,871. Let us check if we can make any of the corner numbers greater than 62,871 by swapping two digits:

• For 16,200: The largest number we can make by swapping two digits is 61,200 (by swapping 1 and 6). But 61,200 is smaller than 62,871. So, 16,200 cannot become a supercell.

This means the four corner cells cannot be the four supercells. This is a very important clue. The problem must have a trick, or the definition of "neighbours" is not all 8 surrounding cells. In some puzzles, "neighbours" only means the 4 cells directly up, down, left, and right. Let us try this definition.

Let us assume "neighbours" means only the 4 cells directly up, down, left, and right.

Let us re-check the original supercell with this new definition:

• 62,871 (center): Its neighbours are 39,344, 23,609, 45,306, 50,319. 62,871 > 39,344 (True) 62,871 > 23,609 (True) 62,871 > 45,306 (True) 62,871 > 50,319 (True) So, 62,871 is still the only supercell.

Now, let us try to find a swap that creates 4 supercells under this 4-neighbour rule. If supercells can be diagonally adjacent, then we can have patterns like the four corner cells. Let's try to make the four corner cells supercells.

• TL (16,200) neighbours: 39,344, 23,609.
• TR (29,765) neighbours: 39,344, 45,306.
• BL (19,381) neighbours: 23,609, 50,319.
• BR (38,408) neighbours: 45,306, 50,319.

We need to swap digits in one number to make 4 supercells. This means the number we swap must be one of the corner cells, and this swap must make it a supercell. Also, the other three corner cells must already be supercells, or become supercells because their neighbours change. But their neighbours do not change.

This is still very difficult. Let's think about which number, if changed, could have the biggest impact. The numbers with small leading digits and large digits later on can become very big. Let's list the maximum possible values by swapping two digits for each number:

• 16,200 -> 61,200 (swap 1 and 6)
• 39,344 -> 93,344 (swap 3 and 9)
• 29,765 -> 92,765 (swap 2 and 9)
• 23,609 -> 93,602 (swap 2 and 9)
• 62,871 -> 82,671 (swap 6 and 8)
• 45,306 -> 65,304 (swap 4 and 6)
• 19,381 -> 91,381 (swap 1 and 9)
• 50,319 -> 90,315 (swap 5 and 9)
• 38,408 -> 83,408 (swap 3 and 8)

Let us try swapping digits in 19,381 to get 91,381 (swapping 1 and 9). This makes 19,381 much larger.

Let's check the grid with 19,381 changed to 91,381. The new grid is: 16,200 | 39,344 | 29,765 23,609 | 62,871 | 45,306 91,381 | 50,319 | 38,408

Now, let us check for supercells (using the 8-neighbour definition, as it is more standard for "all its neighbours"):

1. 16,200: Neighbours are 39,344, 23,609, 62,871, 91,381. It is smaller than all of them. Not a supercell.

2. 39,344: Neighbours include 62,871, 91,381. It is smaller than both. Not a supercell.

3. 29,765: Neighbours include 39,344, 45,306, 62,871, 91,381. It is smaller than all of them. Not a supercell.

4. 23,609: Neighbours include 39,344, 62,871, 91,381, 50,319. It is smaller than all of them. Not a supercell.

5. 62,871: Neighbours include 91,381. It is smaller than 91,381. So, 62,871 is no longer a supercell.

6. 45,306: Neighbours include 62,871, 91,381, 50,319. It is smaller than all of them. Not a supercell.

7. 91,381: Neighbours are 23,609, 62,871, 50,319, 38,408. 91,381 > 23,609 (True) 91,381 > 62,871 (True) 91,381 > 50,319 (True) 91,381 > 38,408 (True) So, 91,381 is a supercell. (This is the 1st supercell).