Zauberspiel (Magic Game)

Description:

The game Zauberspiel (Magic Game) consists of a set of eight numbered cardboard cards that are green on the front. Some of the cards have cutouts, some have single digit numbers printed on the front (2009-25521.jpg), and all but one have long lists of one and two digit numbers written on the reverse side (2009-25520.jpg). German patents No. 140457 and No. 146211 are listed on Card VIII with a mention of foreign patents on the reverse side. Instructions for the game indicate that one person chooses a number between 1 and 100. Card I is placed green side down on a table and the person who has chosen the number is read the numbers from the lists on the backs of Cards II through VIII. For each of the seven cards, the person is asked if his or her number is on the list. If it is on the list, the card is placed green side down on the table. If it is not on the list, the card is rotated 180° before it is placed on the table. The pile of cards is then lifted up and two digits appear next to each other through the cutouts. Unless those two digits are 00, they make up the number chosen; if the digits are 00 then the chosen number is 100. In German patent No. 146211 and in patents issued in the United Kingdom (1903 No. 785, "Improvements in Toy or Amusement Appliances and the like"), and the United States (No. 764,209, "Game of Cards"), the eighth card contains the number 10 where one of the 0s appears on Card VIII of Zauberspiel so that 100 appears if 100 is the chosen number (see images of the patent). This change also causes 102 to appear if 2 is the chosen number.

Zauberspiel was donated to the museum by Eva Landé, neé Morgenroth. In a letter of 1 January 1988 offering the game to the museum, the donor noted that the game had belonged to her mother Gertrud Bejach (Morgenroth) Friedemann and wrote that her "stepfather, Professor Ulrich Friedemann, who played string quartett [sic] with Albert Einstein, showed the game to Einstein who took it home and puzzled over it for days, only to return it unsolved." She also wrote that "The same thing happened with the mathematician [Hermann] Minkowski, who also was an acquaintance of my parents [Gertrud and Julius Morgenroth]." What follows is not a mathematical analysis of the game, but one of closely related games as described in some of the patents.

German Patent No. 146211 was filed in 1902 as an addition to patent No. 140457, "Spiel zur Ermittelung einer beliebigen gedachten Zahl (Game to determine any chosen number)," that had been filed the previous year. Patent No. 140457 describes two games that are closely related to Zauberspiel but involve choosing a number between 1 and 20 rather than 1 and 100. It also indicates the mathematics on which these games are based by noting that all whole numbers can be written as a sum of powers of 2. This statement is equivalent to saying that every whole number can be written a binary number, i.e., as a number written in base 2 as a string of 0s and 1s. The strips are divided into positive (the first ten rows) and negative (the last ten rows) parts. The negative parts are formed by rotating the strips 180° to reverse whether or not there is a cutout at a particular spot on the strip as can be seen in Fig. 1 of the patent (AHB2013q000011.jpg). By identifying a cutout with 0 and a lack of a cutout with 1, the positive parts of the five strips with holes display the binary numbers 00000 through 01001 (0 through 9 in base 10) and are followed in the negative parts displaying the binary numbers 10110 through 11111 (22 through 31 in base 10). The last strip displays the base 10 numbers from which the chosen number must be picked.

A similar analysis can be used for the second game, only three of whose cards are shown in Fig. 2 of the patent (AHB2013q000011.jpg). Here, too, the top half of a card is called the positive part and the bottom half the negative part and is formed from the positive part by reversing whether or not there is a cutout at a spot. The positive part of the four remaining cards can be constructed so that the first card has five cutouts and when the top rows of the next three cards are laid one below another the three-digit binary representations of 0 through 4 (000, 001, 010, 011, 100) can be read from top to bottom.

The major difference between Zauberspiel and the two games described above is that in Zauberspiel only the single digits 0 through 9 appear on cards so that the two digits that appear at the last step will in most cases appear through cutouts of two different sheets. Another difference is that in Zauberspiel only Cards I, II, and III have the bottom half constructed from the top half by reversing whether or not there is a cutout at a given spot. The British patent refers to the placement of the cutouts and base 10 numbers for this game as follows: "The number and arrangement of the figures and perforations in the number sheets a and perforation sheets f is merely determined by empirical rules. … The arrangement of the single numbers on the number sheet a and also on the perforated sheets fmust take place according to definitely prescribed rules" (Patent 1903 No. 785, page 5, lines 10-17). Unfortunately, there is no explanation of the rules needed to place the numbers and, unlike the situation in the previous games, these rules were not discernible simply by examining the binary numbers produced by identifying a cutout with 0 and a lack of a cutout with 1.