2025 AMC 8 Problem 25

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To help you master 2025 AMC 8 Problem 25, this page includes:

The original AMC problem to try, plus the correct answer

Clear, step-by-step solutions using different methods

Key ideas, useful formulas, and takeaways you can apply to similar problems

Related problems from past AMC exams, plus new practice problems that build the same skills

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2025 AMC8 Problem 25

Solution

Solution 1

Global Symmetry Method

Every path has a mirror image. The area on the right side of a path plus the area on the right side of its mirror image exactly equals the area of the entire grid.

The number of paths to travel $a$ steps in one direction and $b$ steps in another direction is given by the combinations formula $(a a + b)$ .

Step 1: Calculate the total number of possible paths

To go from the bottom vertex to the top vertex of the $5 \times 5$ diamond grid, any valid path must take exactly 5 Northeast (NE) steps and 5 Northwest (NW) steps, in some order. We need to choose 5 positions for the NE steps out of the 10 total steps. We use the combinations formula.

Total Paths = (5 10) = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252

Step 2: Understand the grid area and its two halves

The $5 \times 5$ grid contains exactly 25 small squares. Any unbroken path from bottom to top neatly slices the grid into two distinct regions: a left side and a right side. Therefore, for any single path, the area of its left side plus the area of its right side must equal 25.

Area_{left} + Area_{right} = 25

Step 3: Pair each path with its mirror image

Imagine drawing a vertical line straight down the middle of the diamond. If you reflect any path across this line, every NE step flips to a NW step, and every NW step flips to a NE step. This creates a new, valid “mirror image” path.

Step 4: Combine the right-side areas of the paired paths

When we flip a path horizontally, the squares that were sitting on its left side are flipped over to become the right side of the new mirror-image path. Because of this, the right-side area of the mirror path is exactly equal to the left-side area of the original path. When we add the right-side area of a path to the right-side area of its mirror image, we get the total area of the grid.

Area (Path) + Area (Mirror Path) = 25

Step 5: Calculate the final sum across all pairs

We know there are 252 total paths. Since we are pairing them up two-by-two, we can divide by 2 to find the number of pairs. Each pair contributes a combined area of exactly 25 to our final sum. We multiply the number of pairs by 25 to get the answer.

Number of Pairs = \frac{252}{2} = 126

Total Area Sum = 126 \times 25 = 3150

Correct answer: B. $3150$
The total sum of the areas is

3150

, which corresponds to choice B. We can quickly verify this conceptually: by symmetry, the ‘average’ path splits the grid perfectly in half, having a right-side area of 12.5. Multiplying the average area (12.5) by the total number of paths (252) gives 3150.

Solution 2

Local Expectation Method — Square by Square

Instead of looking at whole paths and counting their squares, we can look at each individual square and count how many paths include that square on their right side.

Step 1: Change perspective from paths to squares

We already know there are 252 total paths. Rather than calculating the area path-by-path, we will examine the grid square-by-square. We want to find out how many times each of the 25 squares is shaded (meaning it falls on the right side of a path).

Step 2: Group the squares by symmetry

The vertical axis from the bottom to the top of the diamond divides the 25 squares into three groups: 10 squares on the right half, 10 mirrored squares on the left half, and 5 squares perfectly in the middle.

Step 3: Calculate the shaded count for the 10 paired squares

Take any square on the right half and its mirror image on the left half. For every path that leaves the right square shaded, its reflected mirror path leaves the left square unshaded. Because paths match up perfectly, if you add the number of times the right square is shaded to the number of times the left square is shaded, the total must equal the total number of paths. There are 10 such pairs.

Total shaded per pair = 252

Total for all pairs = 10 \times 252 = 2520

Step 4: Calculate the shaded count for the 5 middle squares

A square resting exactly on the middle axis is its own mirror image. Because the grid and movements are completely symmetrical left-to-right, a middle square must be shaded exactly half of the time. There are 5 such squares.

Times shaded per middle square = \frac{252}{2} = 126

Total for middle squares = 5 \times 126 = 630

Step 5: Combine to find the grand total

We add the total number of shaded squares from the left/right pairs and the middle axis to find the grand total over all paths.

2520 + 630 = 3150

Correct answer: B. $3150$
The total is

3150

. The fact that two completely different viewpoints (counting by paths vs counting by squares) yield the exact same total confirms the logic is sound.

Adapted Problems to Practice

1 / 10

What Formulas Do You Need for This Problem

Combinations for Grid Paths:

(a a + b)

a and b are the total number of steps required in each of the two allowed directions. The path always requires exactly 5 Northeast steps and 5 Northwest steps, making the total number of paths 10 choose 5.

Complementary Pairing:

When a system exhibits perfect symmetry, pair asymmetric objects with their mirror images to find a constant, predictable sum.

Requires a well-defined reflection or operation that maps every valid state to another valid state. The left and right movements are symmetrical, ensuring every path has a partner such that their combined areas sum to the constant total area of the grid.

Key Takeaway: Remember This Strategy for Next Time

A problem asking you to find a large sum over many different arrangements or paths in a symmetric shape.

Try: Look for a way to pair each arrangement with its opposite or mirror image (Complementary Pairing).

Why: Adding the properties of a pair usually yields a constant total. Instead of performing hundreds of calculations, you only need to multiply the constant total by the number of pairs.

A problem asking for the total sum of multiple overlapping regions, areas, or counts.

Try: Change your perspective from 'area per path' to 'paths per square'.

Why: Calculating how often one specific small piece is included across all scenarios is frequently much easier than calculating the size of the whole region for each individual scenario.

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2025 AMC 8 Problem 25

In This Page:

2025 AMC8 Problem 25