top right, with tiles “14” and “15” switched. Loyd
offered a $1,000 prize to the first person submitting a
correct solution.
It turns out, as Loyd well knew, that this puzzle is
impossible to solve. The theory of even and odd
PER
-
MUTATION
s explains why.
Following the snaked path indicated above,
and ignoring the location of the blank cell, one
can record any arrangement of tiles as a list of
the numbers 1 through 15. For instance, the
initial arrangement of tiles appears as the list:
123487659101112151413
Since a larger number appears to the left of a
smaller number nine times, this list represents an
odd permutation of the numbers 1 through 15.
Notice that if one slides a tile horizontally dur-
ing the play of the game, the corresponding list
representing the arrangement of tiles does not
change. If, on the other hand, one slides a tile
vertically, the number of that tile shifts an even
number of places up or down the list that repre-
sents the arrangement of tiles. Shifting a number
one place to the left or right can be effected by a
single transposition, two places to the left or
right can be effected by two transpositions, and
so forth. Thus the result of shifting a number an
even number of places to the left or right is the
result of applying an even number of transposi-
tions, thereby preserving the evenness or oddness
of the arrangement of tiles. In playing the
game, then—given that the initial arrangement
of tiles corresponds to an odd permutation—the
arrangement of tiles will forever remain an odd
permutation. It is impossible, then, to obtain the
arrangement called for by the goal of the puzzle,
namely, 1 234876591011121415
13, an even permutation.
Mathematicians have shown that all arrangements
of tiles that correspond to odd permutations of the
numbers 1 through 15 can in fact be achieved through
the play of the game.
slide rule One can perform simple additions with the
aid of two ordinary rulers. To compute 2.7 + 3.5, for
example, place the end (position “0”) of one ruler at
the location 2.7 along the second ruler. Then read 3.5
units along the first ruler. The corresponding label on
the second ruler is then the desired sum 6.2.
Scottish mathematician J
OHN
N
APIER
(1550–1617)
discovered
LOGARITHMS
near the turn of the 17th cen-
tury. These functions have the remarkable property of
converting computations of multiplication into simpler
computations of addition: log(N×M) = logN+ logM.
In 1622, English mathematician W
ILLIAM
O
UGHTRED
(1574–1660) realized that two sliding rulers, with
labels placed in
LOGARITHMIC SCALE
, will physically
perform the addition of logarithms, and thus allow one
to simply “read off” the result of any desired multipli-
cation. (As a very simple example, imagine we wished
to compute 100 ×1,000 with a pair of base-10 loga-
rithmic rulers. Note then that the mark with label 100
is placed 2 in. along the ruler, and the mark labeled
1,000 3 in. along the ruler. The sum 2 + 3 is, of course,
5 in. along the ruler. But this fifth position is labeled
10,000, which is indeed the product 100 ×1,000.)
Oughtred’s mechanical device of two sliding rulers
is called a slide rule. The device was inspired by the
work of English scholar Edmund Gunter (1581–1626),
who had used a single ruler and a pair of pointers to
accomplish the same feat.
Slide rules were popular up until the 1970s before
the advent of the pocket calculator.
See also N
APIER
’
S BONES
.
slope (grade, gradient) The slope of a line is a mea-
sure of its steepness. This can be determined a number
of ways:
1. Numerically: Two quantities are said to be linearly
related if a unit change in one quantity produces a
constant change in the other. The value of that con-
stant change is the slope of the relationship. For
example, the following table shows the profit made
for a company that sells widgets.
466 slide rule
Slide fifteen puzzle