Discrete and Computational Geometry

We consider a device with rectangular base having no gradations. We show that the number of directly measurable amounts of liquid using the device with its vertices as markers is always 13, independent of its shape. Then we show how the device can measure any integral amount of liquid between 1 and 858 liters. 1 Universal Measuring Devices A common device used to measure liquid in many Japanese stores some years back, was a measuring box with a square base of area 6 and height 1. By tilting the box, and using its vertices as markers, it is possible to keep 6, 3, and 1 liters. The box would have no extra gradations to measure 2, 4, and 5 liters. A store would keep a container holding large amounts of a certain liquid. If a customer wanted to buy a certain integral amount of liquid between 1 and 6, the store owner would proceed as follows: 1. He would immerse his measuring box into the store’s container, and fill it just once. 2. Then he would alternately pour out a certain amount of liquid into the store’s container and the customer’s container. We are interested in studying measuring devices without gradations which nevertheless can be used for measuring any integral amount of liquid up to their full capacity by using the procedure described above. We call such measuring devices universal measuring devices. In the previous works [1,2], we have determined that the largest capacity of a universal measuring device with triangular base is 41. In this work, we consider devices with rectangular bases. Although we include some discussion of this case in the previous work [1], we used the horizontal plane to keep liquid in the device. In this present work, we refer back to the Japanese store owners practice of using only vertices of the device as markers. In particular, we use at least three vertices of the device to specify the plane formed by the surface of the water. J. Akiyama and M. Kano (Eds.): JCDCG 2002, LNCS 2866, pp. 1–8, 2003. c © Springer-Verlag Berlin Heidelberg 2003 2 Jin Akiyama, Hiroshi Fukuda, and Gisaku Nakamura 2 Devices with a Rectangular Base We consider a universal measuring device with rectangular base of area 6. We first find all the amounts of liquid that can be directly measured with the device. We assume that the vertices of these devices are labeled ai, ai for i = 1, 2, 3, 4, as shown in Figure 1. 1 a 2 a 3 a 4 a 1 ' a 2 ' a 3 ' a 4 ' a