1989 AHSME Problems/Problem 22
Problem
A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic,wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks in the set different from the 'plastic medium red circle' in exactly 2 ways? (The 'wood medium red square' is such a block)
(A) 29 (B) 39 (C) 48 (D) 56 (E) 62
Solution
The blocks can be sorted into two identical 
cuboids, one wood and the other plastic, so that in each cuboid the z axis represents the size, the x axis the color, and the y axis the shape.
Suppose the reference block is in position 
in the plastic cuboid.
The wooden blocks already differ from the reference block in one way, so if in the wooden cuboid represents 'wood medium red circle' then any wooden block with two zero coordinates satisfies the requirements. These form the edges of the cuboid which adjoin , so there are 
.
The required plastic blocks must have two non-zero coordinates, so we count the blocks on the three faces around but not on the adjoining edges. There are 
.
In total there are 
blocks meeting the requirements.
![[asy] import three; currentprojection=perspective((7.5,5,5),up=Z); currentlight=nolight; viewportmargin=(1mm,1mm); real c=.5; size(8cm,0); draw((0,0,0)--(0,0,4),Arrow3); draw((0,0,0)--(0,5,0),Arrow3); draw((0,0,0)--(5,0,0),Arrow3); for(int z=0 ; z<3 ; ++z) { for(int x=0 ; x<4 ; ++x) { for(int y=0 ; y<4 ; ++y) { if (((x==0)&&(y*z!=0))||((y==0)&&(x*z!=0))||((z==0)&&(x*y!=0))) {c=.5;} else {c=.75;} if (x*y*z>0) {c=1;} draw(shift(x,y,z)*unitcube,gray(c)+opacity(.6),.5bp+black); }}} [/asy]](http://latex.artofproblemsolving.com/8/9/c/89cd4456781bbd83d58cdf20cf64bc7ffc356210.png)
