Find the total number of cubes in the given figure?

Concept: A solid triangular pyramid (tetrahedron) built from unit cubes has n stacked triangular layers. Counting from the apex, layer k is a triangular arrangement of side k, containing Tk = k(k+1)/2 cubes. The total number of cubes in an n-layer pyramid is the sum of these triangular numbers, which equals the tetrahedral number: Total = T1 + T2 + ... + Tn = n(n+1)(n+2)/6.

Application: The figure has 6 rows of cube-faces from apex to base, so it is a 6-layer triangular pyramid (n = 6). Counting cubes layer by layer, from the apex down:

1. Layer 1 (apex), side 1: T1 = 1 cube

2. Layer 2, side 2: T2 = 1 + 2 = 3 cubes

3. Layer 3, side 3: T3 = 1 + 2 + 3 = 6 cubes

4. Layer 4, side 4: T4 = 1 + 2 + 3 + 4 = 10 cubes

5. Layer 5, side 5: T5 = 1 + 2 + 3 + 4 + 5 = 15 cubes

6. Layer 6 (base), side 6: T6 = 1 + 2 + 3 + 4 + 5 + 6 = 21 cubes

7. Total = 1 + 3 + 6 + 10 + 15 + 21 = 56 cubes

Cross-check: Using the tetrahedral-number formula directly, Total = n(n+1)(n+2)/6 = (6 x 7 x 8)/6 = 56, confirming the layer-by-layer sum. An independent diagonal count also agrees: (6x1) + (5x2) + (4x3) + (3x4) + (2x5) + (1x6) = 6 + 10 + 12 + 12 + 10 + 6 = 56.

So the figure contains 56 cubes in total.