Common developments of three incongruent boxes of area 30

Abstract

We investigate common developments that can fold into plural incongruent orthogonal boxes. Recently, it was shown that there are infinitely many orthogonal polygons that fold into three boxes of different size. However, the smallest one that folds into three boxes consists of 532 unit squares. From the necessary condition, the smallest possible surface area that can fold into two boxes is 22, and the smallest possible surface area for three different boxes is 46. For the area 22, it has been shown that there are 2,263 common developments of two boxes by exhaustive search. However, the area 46 is too huge to search. In this paper, we focus on the polygons of area 30, which is the second smallest area of two boxes that admits to fold into two boxes of size 1×1×7 and 1×3×3. Moreover, when we fold along diagonal lines of rectangles of size 1×2, this area 30 may admit to fold into a box of size 5×5×5. The results are summarized as follows. There exist 1,080 common developments of two boxes of size 1×1×7 and 1×3×3. Among them, there are nine common developments of three boxes of size 1×1×7, 1×3×3, and 5×5×5. Interestingly, one of nine such polygons folds into three different boxes 1×1×7, 1×3×3, and 5×5×5 in four different ways.