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Problem 631

AMC 10/12, early questions

Combinatorics Difficulty 3.8 Multiple choice CEMC Pascal · Canada · 2021

Azmi has four blocks, each in the shape of a rectangular prism and each with dimensions $2\times 3\times 6$ . She carefully stacks these four blocks on a flat table to form a tower that is four blocks high. The number of possible heights for this tower is

Pick one

Official solution

The height of each block is 2, 3 or 6.

Thus, the total height of the tower of four blocks is the sum of the four heights, each of which equals 2, 3 or 6.

If 4 blocks have height 6, the total height equals $4 \times 6 = 24$ .

If 3 blocks have height 6, the fourth block has height 3 or 2.

Therefore, the possible heights are $3 \times 6 + 3 = 21$ and $3 \times 6 + 2 = 20$ .

If 2 blocks have height 6, the third and fourth blocks have height 3 or 2.

Therefore, the possible heights are $2 \times 6 + 3 + 3 = 18$ and $2 \times 6 + 3 + 2 = 17$ and $2 \times 6 + 2 + 2 = 16$ .

If 1 block has height 6, the second, third and fourth blocks have height 3 or 2.

Therefore, the possible heights are $6 + 3 + 3 + 3= 15$ and $6 + 3 + 3 + 2 = 14$ and $6 + 3 + 2 + 2 = 13$ and $6 + 2 + 2 + 2 = 12$ .

If no blocks have height 6, the possible heights are $3 + 3 + 3 + 3 = 12$ and $3 + 3 + 3 + 2 = 11$ and $3 + 3 + 2 + 2 = 10$ and $3 + 2 + 2 + 2 = 9$ and $2 + 2 + 2 + 2 = 8$ .

The possible heights are thus $24, 21, 20, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8$ .

There are 14 possible heights.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.