Dice in a box

In a box in the shape of a cuboid, four kinds of dice are stored in four layers. In the first layer there are dice with an edge length of 12 cm. In each subsequent layer the length of the edge of the die is 2 cm smaller than the length of the edge of the die in the preceding layer. Assuming that between the walls of the box and the dice, as well as between the dice themselves, there are no gaps, calculate:
1. what are the smallest possible internal dimensions of the box
2. how many dice of individual kinds are in this smallest possible box

Final Answer:

a =  120 cm
b =  120 cm
c =  36 cm
n₁ =  100
n₂ =  144
n₃ =  225
n₄ =  400

Step-by-step explanation:

$b=a=120=120\text{cm}$

$c=x_1+x_2+x_3+x_4=12+10+8+6=36\text{cm}$

$n_1={\frac{a}{x_1}}^2={\frac{120}{12}}^2=100$

$n_2={\frac{a}{x_2}}^2={\frac{120}{10}}^2=144$

$n_3={\frac{a}{x_3}}^2={\frac{120}{8}}^2=225$

$n_4={\frac{a}{x_4}}^2={\frac{120}{6}}^2=400$

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