Henry Ernest Dudeney/Modern Puzzles/142 - Economy in String/Solution/Proof 1

by : $142$

 * Economy in String

Solution
$2$ feet by $1$ foot by $\tfrac 2 3$ feet.

Proof
From the general solution:

Thus it follows that the total length of string that passes along the length, breadth or depth must in every case be the same to allow of the maximum dimensions, that is: $4$ feet.

That is:
 * $a x = b y = c z = \dfrac m 3$

The string passes $2$ times along length, $4$ times along breadth, and $6$ times along depth.

Therefore $4$ feet divided by $2$, $4$ and $6$ feet will give us $2$ feet, $1$ foot, and $\tfrac 2 3$ feet respectively for the length, breadth or depth of the largest possible parcel.