This is a question posed to my brother in Grade 5.
- What would be the general approach to solve-
How many cuboids of dimensions $a*b*c$ are there in a cuboid of dimension $d*e*f$?
My brothers approach:
- Just get the ratio $\frac{d*e*f}{a*b*c}$ and round it off.
My approach:
- Check 1: Each of $d$, $e$, $f$ must be larger than $a$, $b$, $c$
- Check 2: Each of $d$, $e$, $f$ must be divisible by $a$ or $b$ or $c$. That is $\frac{a}{d}=0$, $\frac{b}{e}=0$, $\frac{c}{f}=0$.
So my answer would be: $$\frac{24\times 30\times 28}{3\times 5\times 2}=672$$
Your brother's approach is more nearly correct: since there is no requirement that all the rectangular blocks should be cut out in the same orientation, the only obvious upper bound is that the volume of the little blocks should not exceed the volume of the big block. That upper bound is $$ \frac{24\cdot 30 \cdot 29}{5\cdot 3 \cdot 2}=696. $$
You still need to show that that upper bound can be achieved, though. So consider first cutting the $24\times 30 \times 29$ block into a $24\times 30\times 25$ block plus a $24\times 30 \times 4$ block. The first block can be cut into small blocks with no waste because $2|24$, $3|30$, and $5|25$. The second can also be cut with no waste because $3|24$, $5|30$, and $2|4$. So the correct answer is in fact $696$.