Mathematical Analysis of the Pile

To find the number of layers, we first need to identify the mathematical pattern describing the number of balls in each layer.

1. Identifying the Pattern

The number of balls in each layer follows a sequence:

• Layer 1: 1 ball
• Layer 2: 3 balls (1 + 2)
• Layer 3: 6 balls (1 + 2 + 3)
• Layer 4: 10 balls (1 + 2 + 3 + 4)

This sequence consists of triangular numbers. The number of balls in the k-th layer is given by the formula: $T_k = \frac{k(k+1)}{2}$.

2. Formulating the Total

The total number of balls in a pile with 'n' layers is the sum of the first 'n' triangular numbers. The formula for this sum is:

Total Balls = $\frac{n(n+1)(n+2)}{6}$

3. Setting up and Solving the Equation

We are given that the total number of balls is 8436. We can set up the equation:

$\frac{n(n+1)(n+2)}{6} = 8436$

Multiplying both sides by 6, we get:

$n(n+1)(n+2) = 8436 \times 6 = 50616$

We are looking for three consecutive integers whose product is 50616. Instead of solving the cubic equation algebraically, we can test the given options.

4. Testing the Options

A quick way to check is by looking at the last digit of the product for each option:

• For n=34: The product is 34 × 35 × 36. The last digit would be (4 × 5 × 6) → (20 × 6) → 0. This does not match 50616.
• For n=38: The product is 38 × 39 × 40. The last digit would be 0. This does not match 50616.
• For n=32: The product is 32 × 33 × 34. The last digit would be (2 × 3 × 4) → 24 → 4. This does not match 50616.
• For n=36: The product is 36 × 37 × 38. The last digit would be (6 × 7 × 8) → (42 × 8) → 16 → 6. This matches the last digit of 50616.

Let's verify the full calculation for n=36:

36 × 37 × 38 = 1332 × 38 = 50616.

The calculation is correct.

Conclusion

The number of horizontal layers in the pile is 36.