Evening star. [volume] (Washington, D.C.) 1854-1972, 1880-03-27
rho Game of Fifteen.

Editor SARA great deal has been written
on 1ns o a aed a ara
some very interesting points connected with 1t,
desire to present them to your readers.
sented by other correspondents in another form.
but think the solutions that have hitherto
been given have not been stated quite definitely
enough.
propose to consider under what conditions
the problem is solvable, and how it can be done,
and indeed " neither 1t cannot be done In every
Guilty few terms, and in the first case it is
clearly possible 10 arrange them In the original
order, II they have only been misplaced by Or
binary n,Opes in the box. We shall then take no
further notice of this case; again, If one OF the
blocks should be taken out of its place and
jumped over any number Of blocks, we call that
q Jump: II one block is taken out of its place.
put In the place of another, and that one put
back In the place OF the first moved this Is an
tnte1 change, any numter Of noces are lever
considered as anything, but If the blocks were
arranged properly at first, and moved anyhow
and at any stage OF the process any jump or
number of t,'aps should be made, and the sum
of the number OF blocks jumped over is eren.
the problem is always sovable in exactly the
ssme order as the original position. but II the
problem is quite different; this can be reduced
to the inevitable 10, 10 ID, or an interchange of
any two blocks.
II the blocks are placed at random they can
think altogether the easiest and most syste-
matte way OF doing it Is, by first getting 1 in
place, tBen moving uutl IL comes next 10 IL:
then, however, these may be moved around do
not separate them. 1t will then not be difficult
to move and then A. but do not let anything
cases when you can see that something viii be
gained. When the first four blocks are arranged
together they may be in the wrong part of The
ss ss apS' ros Os nsrs u they INTO m tas
and not be moved any more.
and not again be touched, but after the third
row Is made up the 9 and 10 must be brought
round In the lowest row, the II and IL moved
up to the left SO as to make room to move the
Is, IA and15, which must be done until the 13
comes next to the 8, and then the third row
moved In position; the last row will then al
ways be II, 49, H. IS or u, Is, H. With a little
and always certain; the first case is equivalent
to an eren number OF Jumps and the second one
to an odd number. The only question remains
how are we to dispose Of this odd case one says
that 1t cannot be done, which is true In one
sense of the term; but even this question can
always De resolved by turning the box one
ngureS Will read SidewtSe. 1t can also be done |
by reading the first row from left to right, the
second from right to left, the third from left to
think this is not a solution of the question. |
Another solution, which have never heard |
suggested; VII, l think, fairly meet the DIM
top horizontal row win read /, O. 8, ora, 8;
the next s. II or 8. s. s. 9; the third 8, 9, TO-
10 OF II, 1O, II, ana IL Will be seen by moving
as before that the last row can be made to read |
IL, 49, IA or ID, IA. 15; If the IL and Is are
moved down and the l, Q, 8 moved across, or the |
I, 8, IL moved down, they will all be in proper |
order, right side up, and the box will not have |
10 pc turned, the only difference between this /
and the original arrangement being that the |
first space instead Of the sixteenth is the blank
but the numbers are all consecutive. It seems |
to me that this solution meets this case better |
than any have seen. Every problem reduces |
to this: there is vacant space, and it may be
either O mr. on ror ort 1,ol0
right hand corner There Is one more interest
Ing feature OF the matter, that Is: 1t is always |
possible to tell at the starting which of the two
cases Will be the solution of any proposed prob |
lem 50 as to save the trouble and annoyance OF
possibly working out two Independent prob
LENS. it can easily be proved that an inter
OF jumps, for II one block is to occupy anothers
space 1t Is equivalent to moving the blocks UP
and jumping over it; but this one in moving TO
the others original place makes one less ju7nD. |
fer 1t does not jump over itself; we must then
.neerchdnge the block In the first space with l. |
Lne Dlock In the second space with 2, and so on.
keep track OF the new positions which the blocks |
would occupy, but schedule could be made OF |
IL on piece OF paper, and 1t would not be very |
difficult to trace 1t on the board. After a few |
trials the problem then now could be stated In |
one OF two ways, first If the whole number OF
fnterchaRges In order to bring all the blocks In
proper position Is even, the vacant space IL at
the bottom, II odd the vdcant space is at the top
or secondly. If at any time an int rct1p9e brings
DOtn BLOCKS INTO position, and this happens an
odd number OF times, the vacant space Is at the
bottom, and II an even number at the top this |
arises from the fact that the whole number of
blocks that have to be brought into position is
an ODD number, Is. OF course it must be under
stood that If any of the blocks are originally in
position there are so many less tnterchangss to
be made, and In the second case the number of
blocks originally in position must be added to
those 1Rterchanges which bring both blocks In
position. The position OF the blocks will al
ways be either
8 Q
s1S Il 70 or s,S a.