The allotrope of carbon used for making the black cores of pencils.

Explanation

**Step-by-Step Solution:**

1. **Identify the Allotropes of Carbon**: Carbon has several allotropes, including diamond, graphite, fullerenes, and amorphous carbon.

2. **Understand the Properties of Graphite**: Among these allotropes, graphite is known for its layered structure. Each layer consists of carbon atoms arranged in a hexagonal pattern.

3. **Relate Graphite to Pencil Cores**: The black cores of pencils are made from a mixture of graphite and clay. The graphite provides the black color and allows the pencil to write smoothly.

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