The second diagram will serve to show why the first cigar must be placed on end. (And here I will say that the first cigar that I selected from a box I was able so to stand on end, and I am allowed to assume that all the other cigars would do the same.) If the first cigar were placed on its side, as at F, then the second player could place a cigar as at G—­as near as possible, but not actually touching F. Now, in this position you cannot repeat his play on the opposite side, because the two ends of the cigar are not alike.  It will be seen that GG, when placed on the opposite side in the same relation to the centre, intersects, or lies on top of, F, whereas the cigars are not allowed to touch.  You must therefore put the cigar farther away from the centre, which would result in your having insufficient room between the centre and the bottom left-hand corner to repeat everything that the other player would do between G and the top right-hand corner.  Therefore the result would not be a certain win for the first player.

399.—­THE TROUBLESOME EIGHT.

[Illustration: 

+—–­+—–­+—–­+
| 41/2| 8 | 21/2|
+—–­+—–­+—–­+
| 3 | 5 | 7 |
+—–­+—–­+—–­+
| 71/2| 2 | 51/2|
+—–­+—–­+—–­+

]

The conditions were to place a different number in each of the nine cells so that the three rows, three columns, and two diagonals should each add up 15.  Probably the reader at first set himself an impossible task through reading into these conditions something which is not there—­a common error in puzzle-solving.  If I had said “a different figure,” instead of “a different number,” it would have been quite impossible with the 8 placed anywhere but in a corner.  And it would have been equally impossible if I had said “a different whole number.”  But a number may, of course, be fractional, and therein lies the secret of the puzzle.  The arrangement shown in the figure will be found to comply exactly with the conditions:  all the numbers are different, and the square adds up 15 in all the required eight ways.

400.—­THE MAGIC STRIPS.

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.