Question

An artist stacks $14$ cubes, each with edges of $1$ meter, into a staircase: a $3 \times 3$ block of $9$ cubes rests on the ground, a $2 \times 2$ block of $4$ cubes sits on top of it flush into one back corner, and a single cube sits on top of that at the same corner. She paints the entire exposed surface of the sculpture, meaning every face except those resting on the ground. How many square meters does she paint?

$21$

$24$

$33$

$37$

$42$

Solution:

Because each higher layer is flush into one corner, each of the four vertical sides shows a stepped profile of $3 + 2 + 1 = 6$ exposed square faces, for $4 \times 6 = 24$ side faces.

Viewed from directly above, the top faces cover the full $3 \times 3$ footprint, adding $9$ more faces. The bottom rests on the ground and is not painted, so the total is $24 + 9 = 33$ square meters.

Thus, the correct answer is C .

1989 AMC 8 Problem 23

Solution:

Problem 23 in Other Years