Packing 1×2×4 bricks in a cubical box

Abstract

Given a cubical box C 2n+1 of side 2n + 1 and a supply of 1 × 2 × 4 bricks, it is proved that if n ≥ 2, then (A1) one can pack n 3 + 3n 2 +1 2 bricks for n odd, and n 3 + 3n 2 2 bricks for n even, (A2) the capacity of C 2n+1 is ≤ 1 2 n(n + 1)(2n + 1), and if n ≡ 1 or 2 (mod 4), this upper bound for the capacity can be reduced by 1.