Rodney builds increasingly large shapes out of unit rods. The smallest uses six rods and each larger shape adds a layer of hexagons around the previous shape.Starting with 20242024 rods, Rodney builds a single shape like this, as large as possible. How many rods will he have left over?

To solve this problem, we need to understand the pattern of how many rods are needed for each layer of the shape.

1. The first layer (smallest shape) uses 6 rods.
2. The second layer adds a hexagon to each side of the original hexagon, which means it adds 6 more hexagons. Each hexagon requires 6 rods, so the second layer uses 6*6 = 36 rods.
3. The third layer adds a hexagon to each side of the shape from layer 2, which means it adds 12 more hexagons. So, the third layer uses 12*6 = 72 rods.

From this, we can see that the number of rods needed for each layer forms a sequence: 6, 36, 72, and so on. This is an arithmetic sequence where each term increases by 36 rods.

To find out how many layers Rodney can build with 20242024 rods, we need to find the sum of this arithmetic sequence until it exceeds 20242024. The formula for the sum of an arithmetic sequence is:

Sum = n/2 * (a + l)

where: n is the number of terms (layers), a is the first term (6 rods for the first layer), and l is the last term.

We don't know n or l, but we know that the sum must be less than or equal to 20242024. So, we need to keep adding terms to the sequence until we reach or exceed 20242024.

Once we find the largest possible shape Rodney can build, we subtract the total number of rods used from 20242024 to find out how many rods he will have left over.