THE MAGIC STRIPS.
(
Magic Squares Problem.)
I happened to have lying on my table a number of strips of cardboard,
with numbers printed on them from 1 upwards in numerical order. The idea
suddenly came to me, as ideas have a way of unexpectedly coming, to make
a little puzzle of this. I wonder whether many readers will arrive at
the same solution that I did.
Take seven strips of cardboard and lay them together as above. Then
write on each of them the numbers 1, 2, 3, 4, 5, 6, 7, as shown, so that
the numbers shall form seven rows and seven columns.
Now, the puzzle is to cut these strips into the fewest possible pieces
so that they may be placed together and form a magic square, the seven
rows, seven columns, and two diagonals adding up the same number. No
figures may be turned upside down or placed on their sides--that is, all
the strips must lie in their original direction.
Of course you could cut each strip into seven separate pieces, each
piece containing a number, and the puzzle would then be very easy, but I
need hardly say that forty-nine pieces is a long way from being the
fewest possible.
Answer:
There are of course six different places between the seven figures in
which a cut may be made, and the secret lies in keeping one strip intact
and cutting each of the other six in a different place. After the cuts
have been made there are a large number of ways in which the thirteen
pieces may be placed together so as to form a magic square. Here is one
of them:--
[Illustration:
+-------------+
|1 2 3 4 5 6 7|
+---------+---+
|3 4 5 6 7|1 2|
+-----+---+---+
|5 6 7|1 2 3 4|
+-+---+-------+
|7|1 2 3 4 5 6|
+-+---------+-+
|2 3 4 5 6 7|1|
+-------+---+-+
|4 5 6 7|1 2 3|
+---+---+-----+
|6 7|1 2 3 4 5|
+---+---------+
]
The arrangement has some rather interesting features. It will be seen
that the uncut strip is at the top, but it will be found that if the
bottom row of figures be placed at the top the numbers will still form a
magic square, and that every successive removal from the bottom to the
top (carrying the uncut strip stage by stage to the bottom) will produce
the same result. If we imagine the numbers to be on seven complete
_perpendicular_ strips, it will be found that these columns could also
be moved in succession from left to right or from right to left, each
time producing a magic square.
