Brainteasers & Puzzles

The Loosened Cube Shell

Weather strips the outer layer from a 10 by 10 by 10 block of glued unit cubes. Counting the untouched 8 by 8 by 8 core instead of the shell's faces, edges, and corners gives 1000 - 512 = 488 fallen cubes in one subtraction.

solvedeasy1 min

Picture a 10 by 10 by 10 macro-cube built from 1 by 1 by 1 micro-cubes glued together. Weather loosens the entire outermost layer of micro-cubes, which falls to the ground. How many micro-cubes are on the ground?

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#Strip the shell, keep the core

Counting the shell directly means juggling faces, edges, and corners. It is far cleaner to count what stays behind. Removing the outer layer shaves one micro-cube off each end of all three dimensions, so the surviving core is a solid cube of side 10−2=810 - 2 = 8,

83=512.(1)8^3 = 512. \tag{1}

#Count by difference

The whole cube holds 1000 micro-cubes. Strip the outer shell and what remains is a solid 8 by 8 by 8 core of 512, since each dimension loses one cube at each end. The shell on the ground is the difference, 1000 minus 512, which is 488.

The shell is everything the core is not, so

103−83=1000−512=488.(2)10^3 - 8^3 = 1000 - 512 = 488. \tag{2}

A total of 488 micro-cubes hit the ground. The same complementary count works for any side mm, where the shell holds m3−(m−2)3=6m2−12m+8m^3 - (m-2)^3 = 6m^2 - 12m + 8, and 6⋅100−120+8=4886 \cdot 100 - 120 + 8 = 488 confirms the figure.