A 5x5x5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them? [asy]
fill((2,0)--(3,0)--(3,5)--(2,5)--cycle,gray(.7));
fill((0,2)--(0,3)--(5,3)--(5,2)--cycle,gray(.7));
path p=(0,0)--(5,0);

draw(p,linewidth(1));
draw(shift(0,1)*p,linewidth(1));
draw(shift(0,2)*p,linewidth(1));
draw(shift(0,3)*p,linewidth(1));
draw(shift(0,4)*p,linewidth(1));
draw(shift(0,5)*p,linewidth(1));

draw((0,0)--(0,5),linewidth(1));
draw((1,0)--(1,5),linewidth(1));
draw((2,0)--(2,5),linewidth(1));
draw((3,0)--(3,5),linewidth(1));
draw((4,0)--(4,5),linewidth(1));
draw((5,0)--(5,5),linewidth(1));

[/asy]
The number of painted faces is $9(6)=54$.  However, the four edge faces painted on each face are also painted on the other face.  Thus $4(6)=24$ of the painted faces are on cubes with two painted faces.  These account for only $12$ painted cubes, in addition to the $54-24=30$ singly painted cubes.  So there are $42$ painted cubes, leaving $125-42=\boxed{83}$ unpainted cubes.
83
Level 5
Counting & Probability
Problem: A 5x5x5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them? [asy]
fill((2,0)--(3,0)--(3,5)--(2,5)--cycle,gray(.7));
fill((0,2)--(0,3)--(5,3)--(5,2)--cycle,gray(.7));
path p=(0,0)--(5,0);

draw(p,linewidth(1));
draw(shift(0,1)*p,linewidth(1));
draw(shift(0,2)*p,linewidth(1));
draw(shift(0,3)*p,linewidth(1));
draw(shift(0,4)*p,linewidth(1));
draw(shift(0,5)*p,linewidth(1));

draw((0,0)--(0,5),linewidth(1));
draw((1,0)--(1,5),linewidth(1));
draw((2,0)--(2,5),linewidth(1));
draw((3,0)--(3,5),linewidth(1));
draw((4,0)--(4,5),linewidth(1));
draw((5,0)--(5,5),linewidth(1));

[/asy]
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many unit cubes remain unpainted, first calculate the total number of unit cubes in the 5x5x5 cube, which is 125. Next, analyze the painting pattern on each face of the cube. Each face has a 5x5 grid, and nine unit squares are painted. The painted squares form a cross pattern, covering the center square and the four squares adjacent to it, plus the center square of the adjacent rows and columns.

Each face has 9 painted squares. Since the cube has 6 faces, the initial calculation suggests 6 faces × 9 squares = 54 painted squares. However, this count includes overlaps where the same unit cubes are painted on multiple faces. 

Identify the unit cubes that are painted on multiple faces. The corners of the cube are painted on three faces, and the edges (excluding corners) are painted on two faces. Each corner cube contributes to three painted squares, and there are 8 corners. Each edge has 3 unit cubes (excluding corners), and there are 12 edges, contributing to 2 painted squares each.

Calculate the total number of unique painted unit cubes. The center cube of each face is painted once, and there are 6 such cubes. The cubes on the edges and corners must be accounted for to avoid double counting. 

After determining the total number of unique painted unit cubes, subtract this from the total of 125 unit cubes. The result gives the number of unpainted unit cubes. 

Following this reasoning, the final count of unpainted unit cubes is 83.<|END_OF_LATENT|> The final answer is $83$.