Labled Viking Longship

[The Problem of the Week] full, half full and empty

The problem of the week for apprentices:

Three brothers received 21 identical sealed bottles with different amounts of an orange soda: 7 were full, others 7, half-full, and the remaining 7, empty.

How divided the 21 bottles so that each receives the same number of bottles and drink the same amount of unopened bottles?

The solution, of course! Below the photo.

[In the picture you can see a three-dimensional model Klein bottle, a closed surface has no interior and exterior, and is equivalent in area closed to the Möbius strip . We, in our three dimensional reality, we are used to closed surfaces, like a soccer ball, a closed bottle, an unopened box, with interior and exterior, and can not be passed from one to another without breaking or going through the area. In reality, the Klein bottle is a three dimensional object, but four-dimensional (the fourth dimension), a surface that does not cut itself, but as shown in the picture, to represent in three dimensions is needed that " neck "of the bottle finish inside the" belly "to join the fund, through the surface, something that would not happen in four dimensions. The picture seems very nice: note the absurd scale of measurement, in which the volume content is always zero (in fact, having no inside or outside, anything that is "meta" in the bottle can not be isolated abroad, which this bottle is good for "saving" anything) also note at the bottom the English phrase Imported from the 4th Dimension, "imported from the fourth dimension." The image was taken from this site ]

Solution:

We first realize that in total there are 21 bottles, and soda are 7 whole bottles and 7 half bottles, which are 10 ½ bottles, thus spreading among the three, each must have 7 bottles, and soda 3 ½ bottles each. There are at least two different solutions, as anyone who thinks a little can be discovered, and I think they are the only two solutions.

One solution might be:

-the first brother, 3 full bottles, 1 bottle half full, and 3 empty.

, the second brother, 3 full bottles, 1 bottle half full, and 3 empty.

, the third brother, 1 bottle full, 5 half-filled bottles, and 1 empty.

A second solution could be:

-the first brother, 2 full bottles, 3 bottles half-full, and 2 empty.

, the second brother, 2 full bottles, 3 bottles half-full, and 2 empty.

, the third brother, 3 full bottles, 1 bottle half full, and 3 empty.

Note: This issue has been taken from the textbook publisher 3 º ESO SM.