SOLITAIRE. 735 



2nd Series. 2, 4, 2 ... (2) even. 



-^rd Series. 3, 5, 10, 12, 3 . . (4) even. 

 4th Series. 9, 13, 14, 9. . . (3) uneven. 



We have now four series, the total number of points equal 15, as there 

 ought to be, for one cube is absent. 



Let us now take another example (see fig. 866), and by working as 

 before we have four series again, viz : 



1st Series. I, 7, I ... (2) even. 



2nd Series. 2, n, 3, 8, 4, 15, 2. . (6) even. 



^rd Series. 5, 12, 13, 5 . , (3) uneven. 



^th Series. 9, 14, 10, 9 . . (3) uneven. 



This gives us only 14 as a total, because 6 has not been touched at all. 



And now for the rule, so that we may be able to ascertain in advance, 

 when we have established our series, whether we shall find our puzzle right 

 or wrong at the end. We must put aside all unplaced numbers and take no 

 notice of uneven series. Only the even series must be regarded. 



Thus if we do not find i, or if we find 2, 4, or 6, the problem will come 

 into A as a result. If we find I, 3, 5, or 7, the case will eventuate as in B 

 (fig. " ). Let us apply the rule to the problems we have worked, and then 

 the reason will be apparent. 



In the first we find three even series; the problem will then end as in 

 B diagram (fig. 864), for the number of like series is odd. 



In the second we find two even series (pairs) ; we shall find our problem 

 work out as in diagram A (fig. 864), for the number of like series is even, 

 one pair in each. 



We are now in possession of a simple rule, both rapid and infallible, 

 and which will save considerable trouble, as we can always tell beforehand 

 how our puzzle will come out. Any one can test the practicability of the 

 rules for himself, but we may warn the reader that he will never be able to 

 verify every possible instance, for the possible cases are represented by the 

 following sum 



2x3x4x5x6x7x8x9x10x11x12x13x14x1 5. 

 That is to say, 1,307,674,368,000 in all. 



SOLITAIRE. 



This somewhat ancient amusement is well known, and the apparatus 

 consists of a board with holes to receive pegs or cups to receive the balls, as 

 in the illustrations (figs. 867 and 870.) The usual solitaire board contains 

 thirty-seven pegs or balls, but thirty-three can also be played very well. Many 

 scientific people have made quite a study of the game, and have published 



