Transfer puzzle

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Transfer puzzles are a well-known type of brain teaser . In the simplest version, it contains three vessels of known volume, one of which is completely filled with water at the beginning, while the other two are empty. The aim is to measure certain amounts of water. However, since the vessels do not have a scale, the measuring process must be carried out by multiple decanting.

example

The oldest and best-known example of a transfer puzzle is the following task: A vessel that holds 8 liters is filled to the brim with water. There are also two other vessels that hold 3 and 5 liters, both empty. How can you measure exactly 4 liters by repeatedly decanting, i.e. have 4 liters of water each in both the 5 and 8 liter containers?

To solve this, you can note the individual states as triples by specifying how many liters there are in the individual vessels (in the order 3, 5, 8). The initial state is then (0, 0, 8), the desired end state (0, 4, 4).

  • First you fill the 5 liter jar from the 8 liter jar up to the brim, leaving 3 liters: (0, 5, 3)
  • From the 5-liter jar you fill the 3-liter jar to the brim: (3, 2, 3)
  • The 3-liter container is emptied into the 8-liter container: (0, 2, 6)
  • The contents of the 5 liter jar are poured into the 3 liter jar: (2, 0, 6)
  • The 5-liter jar is refilled from the 8-liter jar: (2, 5, 1)
  • Fill the 3-liter jar from the 5-liter jar: (3, 4, 1)
  • The contents of the 3 liter jar are poured back into the 8 liter jar: (0, 4, 4)

history

The first written record of a transfer problem can be found in the Annales Stadenses Chronicle , which Albert von Stade compiled in the 13th century. The entry for the year 1152 has included some puzzles, including the above transfer puzzle. In the 16th century Niccolò Tartaglia dealt with these problems, so that they are often attributed to him. In 1917, the puzzle expert Henry Dudeney wrote that until now such tasks have only been solved with trial and error , but he believes that there are formulas at least for special cases. In 1939 MCK Tweedie gave a systematic solution. In the 20th century, Abraham S. Luchin's tasks of this type were used in psychological experiments.

Mathematical analysis

A parallelogram is embedded in an equilateral triangle, the lower left corner of which coincides with that of the triangle and the upper right corner of which lies on the right edge of the triangle. It is divided into smaller triangles, so that together with the edges of the parallelogram there are 4 horizontal lines and 6 lines parallel to the left edge of the triangle. Several points of intersection are labeled with trilinear coordinates, a red path leads from (0, 0, 8) bottom left to (0, 5, 3) right and further over (3, 2, 3), (0, 2, 6), ( 2, 0, 6), (2, 5, 1) and (3, 4, 1) after (0, 4, 4).
Graphical solution of the above transfer puzzle according to Tweedie

Tweedie used trilinear coordinates to analyze transfer puzzles . The above example can be solved for an equilateral triangle with a height of 8, the fill quantities denote points in this triangle if they are read as trilinear coordinates. Since the first two vessels can only contain 3 or 5 liters, not all points of the triangle correspond to valid filling states; the permitted points form a parallelogram that is contained in the triangle. When decanting, a vessel always remains untouched, so that one moves on parallels to the sides of the triangle. In addition, a vessel must always be completely emptied or filled when decanting, so that the path in the parallelogram always goes to its edge. In fact, the path is already determined by the first step, all further steps arise automatically.

With other vessel sizes, instead of the parallelogram, an irregular pentagon or hexagon can result, but the solution idea remains the same.

The special feature of the path is that the entry and exit angles always match at the edge points. You can imagine it as the path of an ideal billiard ball on a parallelogram-shaped table.

An analogous solution is possible with more than three vessels, the figures are in correspondingly higher-dimensional spaces.

An algebraic solution comes from Paolo Boldi, Massimo Santini and Sebastiano Vigna. This analysis also gives upper and lower bounds for the number of transfers required.

Individual evidence

  1. ↑ Heinrich Hemme : Kopfnuss. 101 math puzzles from four millennia and five continents. Verlag CH Beck , 2012. ISBN 978-3-406-63704-9 . P. 30.
  2. ^ Henry Ernest Dudeney: Amusements in Mathematics. 1917. p. 109. ( Amusements in Mathematics in Project Gutenberg ( currently not available to users from Germany ) )
  3. ↑ MCK Tweedie: A Graphical Method of Solving Tartaglian Measuring Puzzles. In: The Mathematical Gazette. Vol. 23, No. 255, July 1939. pp. 278-282. ( JSTOR 3606420 )
  4. ↑ Alexander Bogomolny: Barycentric coordinates, three jugs application. Retrieved April 29, 2018 .
  5. ^ Paolo Boldi, Massimo Santini and Sebastiano Vigna: Measuring with Jugs. In: Theoretical Computer Science. 282 (2), 2002. pp. 259–270. ( Online , PDF)

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