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Problem 717

AMC 10/12, early questions

Number theory Difficulty 3.3 Multiple choice CEMC Cayley · Canada · 2025

Teddy has $10$ rectangular
blocks each of which measures $\text{3 cm}$ by $\text{4 cm}$ by
$\text{5 cm}$ . He builds a stack
that is exactly $\text{21 cm}$ high,
where each block can be stacked on any of its faces. What is the
smallest number of blocks that Teddy could use to make this stack?

Pick one

Official solution

If Teddy is able to build a stack with height $21\text{ cm}$ using exactly $4$ blocks, then the average height (the
vertical dimension) of each of the blocks is $\dfrac{21\text{ cm}}{4}=5.25\text{ cm}$ .

However, the largest dimension of each block is $5\text{ cm}$ , and so it is not possible to
build a stack with height $21\text{ cm}$ using $4$ (or fewer)
blocks.

If Teddy builds a stack using $4$
blocks positioned so that the vertical dimension of each is $4\text{ cm}$ , and adds $1$ more block to the stack, positioned so
that its vertical dimension is $5\text{ cm}$ , then the stack has height $4\times 4\text{ cm} + 1\times5 \text{ cm}=21\text{ cm}$ .

The smallest number of blocks that Teddy can use to build a stack with
height $21\text{ cm}$ is $5$ .

Can you find two more ways that Teddy can build a $21\text{ cm}$ stack using exactly $5$ blocks?

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