The 15 puzzle in Arno Schmidt’s “Alexander”

Arno Schmidt (1914 - 1979)

Arno Schmidt (1914 – 1979)

In his novela “Alexander or What is Truth” (1949), Arno Schmidt discusses the question of historical truth about Alexander the Great (4th century B.C.). As usual in Schmidt’s works, the story unravels in small paragraphs of first person narration and inner monologue. His alter ego is Lampon of Samos, a 18 year old Greek student of the great Aristotle (who was also the teacher of Alexander), who travels to Babylon at the time when Alexander is dying in the city. Schmidt is notorious for his demanding prose form, his linguistic acrobatics and his mind games inspired from his own vast readings. His encyclopedic texts are highly interactive in the sense that they demand the engagement and participation of the reader and a kind of “homework” to be done in parallel, clarifying the author’s continuous direct references or allusions to Literature, History, Geography, Astronomy and Mathematics. In “Alexander” the author even presents to his readers two specific puzzles, when he mentions, at some point, that Lampon and his friend Hipponax, “out of pure uselessness” exchange challenging problems:

Watching quickly gets to be a bore: Hipponax (who is staying with the baggage this evening, ‘holding position’, as he says in the army jargon of the day), sat down next to me, and out of pure uselessness we exchanged riddles and magic tricks. I made a 16-square field for him, filled them at random with the numbers 1-15 and had him use the open field to shift them back into normal order (he’d never seen that one). In return he explained: ‘The vendor next door has 3 kinds of wine, at 10, 3 and 1/2 drachmas a jug; here are 100 drachmas: bring me 100 jugs,’ and added: ‘It’s not the answer that’s interesting, that’s easy to find. But rather the exact mathematical method for solving it, by way of setting up equations -‘ Well yes, in this heat you can’t always be clever.

The problem suggested by Hipponax is a variation of the well known “100 fowl problem”, traced to Zhang Quijian’s “Mathematical Manual” (a Chinese mathematical text, dating probably from the 5th – 6th century A.D.) and it may be viewed as an anachronism, though it could also be an allusion by Schmidt to a chain of influence from Greek to Indian and Chinese Mathematics. The obvious anachronism is the problem suggested by Lampon, which refers to the “15 puzzle”, a sliding puzzle quite known and popular even to this day. It was believed for some time that it was an invention of the recreational mathematician Sam Loyd (1841 – 1911), though this was based on Loyd’s persistent own claims. The puzzle was in fact invented by some Noyes Palmer Chapman who presented it to friends for the first time in 1874. Lampon similarly presents it to his friend, without mentioning the inventor (though it could be meant by Schmidt to point to Aristotle). Apart from the anachronism, the mind game Schmidt plays here is somewhat concealed: Lampon fills in the fields of the puzzle in random yet the puzzle is actually solvable under a certain condition and not for all random initial arrangements. It can be proved that for any possible arrangement of the 15 puzzle a specific number V constructed from this arrangement is invariant and must be equal to 1. Any arrangement not satisfying this simple requirement is impossible and cannot appear in the puzzle. To understand how V is constructed, one needs first to understand two other numbers related to the 15 puzzle, which we will denote here by R and P:

  • R is the row number of the empty field in the puzzle. Therefore if the empty field is for example in the second row then R=2
  • The aim of the puzzle is to arrange the fields in ascending order 1, 2, 3, 4, 5, … , 15, placing the empty field last.This is the solved arrangement of the puzzle.
    A solvable arrangement of the 15 puzzle, with 19 inversions

    A solvable arrangement of the 15 puzzle, with 19 inversions

    In any unsolved arrangement the fields are not placed in ascending order and a number of inversions is observed, i.e. some fields are preceded by fields marked with bigger numbers. The inversion number of any specific field is the number of the preceding fields marked with bigger numbers than this specific field. For example in the arrangement of the picture, the number of inversions for the field marked with 6 is 4 because there are 4 preceding fields marked with bigger numbers than 6 (the ones marked with 14, 8, 12, 7). The total number P of inversions in the puzzle is the sum of the inversion numbers of all fields (except the empty field). For the arrangement of the picture the total number of inversions is 19.

It can be proved that the sum D of the row number R of the empty field and the total number of inversions P is always an even number (the proof can be found here). We now define the number V=(-1)D For the arrangement of the picture D=R+P=3+19=22. Therefore the number

V=(-1)D

is indeed equal to 1 for any possible arrangement of the 15 puzzle. Though Sam Loyd was not the inventor of the puzzle, he created an unsolvabe variation by exchanging positions of 14 and 15 in the solved arrangement of the puzzle and offered 1000$ to anyone that would solve it. His arrangement changed P from 0 (in the solved arrangement) to 1 (when 14 and 15 exchanged positions) without changing the row number R=4, altering thus the parity of D from even (D=0+4=4, in the solved arrangement) to odd (D=1+4=5, in his arrangement) and resulting to V=-1. Filling in the fields of the puzzle randomly, as Lampon did, produces solvable arrangements only half of the time. Schmidt’s description does not clarify whether this was known to Lampon, and the young Greek was playing a joke upon his friend, or Lampon believed that any arrangement is indeed possible. In fact, it is not clear by the text whether Schmidt himself was aware of impossible arrangements in the puzzle. However, the unsolvability of the puzzle under these certain conditions was proved in 1879 (Johnson and Story, American Journal of Mathematics, Vol.2 No.4, 397 – 404) and Schmidt might have known about it, as he had displayed considerable amateur interest in Mathematics. Therefore it seems quite possible that this could be another little trick, a mind game of Schmidt who, at this specific instance, is presenting Lampon and Hipponax trying to play a game with each other’s mind (neither of the two accomplishing a solution of the problem presented to them) while at the same time he himself is playing a game with his readers’ minds.