It is said that the inhabitants of Montenegro have a little dice game that is both ingenious and well worth investigation.  The two players first select two different pairs of odd numbers (always higher than 3) and then alternately toss three dice.  Whichever first throws the dice so that they add up to one of his selected numbers wins.  If they are both successful in two successive throws it is a draw and they try again.  For example, one player may select 7 and 15 and the other 5 and 13.  Then if the first player throws so that the three dice add up 7 or 15 he wins, unless the second man gets either 5 or 13 on his throw.

The puzzle is to discover which two pairs of numbers should be selected in order to give both players an exactly even chance.

398.—­THE CIGAR PUZZLE.

I once propounded the following puzzle in a London club, and for a considerable period it absorbed the attention of the members.  They could make nothing of it, and considered it quite impossible of solution.  And yet, as I shall show, the answer is remarkably simple.

Two men are seated at a square-topped table.  One places an ordinary cigar (flat at one end, pointed at the other) on the table, then the other does the same, and so on alternately, a condition being that no cigar shall touch another.  Which player should succeed in placing the last cigar, assuming that they each will play in the best possible manner?  The size of the table top and the size of the cigar are not given, but in order to exclude the ridiculous answer that the table might be so diminutive as only to take one cigar, we will say that the table must not be less than 2 feet square and the cigar not more than 41/2 inches long.  With those restrictions you may take any dimensions you like.  Of course we assume that all the cigars are exactly alike in every respect.  Should the first player, or the second player, win?

MAGIC SQUARE PROBLEMS.

    “By magic numbers.” 
    CONGREVE, The Mourning Bride.

This is a very ancient branch of mathematical puzzledom, and it has an immense, though scattered, literature of its own.  In their simple form of consecutive whole numbers arranged in a square so that every column, every row, and each of the two long diagonals shall add up alike, these magic squares offer three main lines of investigation:  Construction, Enumeration, and Classification.  Of recent years many ingenious methods have been devised for the construction of magics, and the law of their formation is so well understood that all the ancient mystery has evaporated and there is no longer any difficulty in making squares of any dimensions.  Almost the last word has been said on this subject.  The question of the enumeration of all the possible squares of a given order stands just where it did over two hundred years ago.  Everybody knows that there is only one solution for