Glued Little Cubes

Algebra Level 2

Joe-Ann has several little 1 × 1 × 1 1 \times 1 \times 1 cubes. She joins the cubes using a drop of glue for each pair of faces in contact to each other. She made a 2 × 2 × 2 2\times 2 \times 2 cube with eight little 1 × 1 × 1 1 \times 1\times 1 cubes and used 12 12 drops of glue. She made a 3 × 3 × 3 3\times 3 \times 3 cube with twenty-seven little 1 × 1 × 1 1 \times 1 \times 1 cubes and used 54 54 drops of glue.

If she wants to make a 10 × 10 × 10 10\times 10 \times 10 cube with a thousand little 1 × 1 × 1 1 \times 1\times 1 cubes how many drops of glue will she use?


The answer is 2700.

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1 solution

Jordan Cahn
Oct 27, 2018

Every face that is not on the surface of the cube gets a drop of glue, and ever drop of glue attaches two faces. Thus we have: total faces exterior faces faces per glue drop = 6 × number of cubes surface area 2 = 6000 600 2 = 2700 \frac{\text{total faces} - \text{exterior faces}}{\text{faces per glue drop}} = \frac{6\times\text{number of cubes} - \text{surface area}}{2} = \frac{6000 - 600}{2} = \boxed{2700}

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