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NOTES AND QUERIES. 1 12 s.vi. APRILS, im

when the knight is on any square of one of the diamond systems he cannot pass to a square on the other diamond system : similarly, when on a square of one of the square systems, he cannot pass to a square on the other square system. He can only pass from a diamond to a square system, and from a square to a diamond system. It follows therefore that if the knight starts from any square in a diamond system to end his tour on any unprescribed square of a different colour, the sequence must take this order : diamond, square, diamond, square. Similarly, if he starts from any square in a square system, the order must be square, diamond, square, diamond. Dr. Roget's method in this case is to complete each system of 16 " halts " before passing on to the next system.

It is interesting to observe how the figure of the cross dominates the arena. This is especially apparent when counters of four different colours to mark the " halts " of the knight are used. Dividing the keyboard into four quarters, it will be at once seen that in each quarter the eight square squares have assumed the cross form ; likewise the eight diamond squares ; that the two diamonds cut crossway throvigh the cross formed squares ; finally that there is a cross at the centre of the board composed of a portion of the two diamond systems.

There are two classes of tours : the un- prescribed terminal and the prescribed terminal. In either case the starting square is optional ; but in the class of a prescribed terminal, this must be of a different colour from that of the terminal ; so only optional as to 32 squares. The following is an example of an unprescribed terminal tour, which

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quite unintentionally on my part furnishes- an example of the re-entering type of tours, the starting and the terminal squares being one move apart. Such a tour gives rise to an interminable network. From a tour of this type it is maintained by one French author and student of the game that no fewer than- 128 variants could be accomplished.

The following is an example of a prescribed terminal tour, which also consists of two- classes : one when the terminal square is in the same system as the starting square ; the other when it is in any of the other three systems. We will take the latter first. The starting square is in a diamond system, the terminal in a square. Proceed as follows r Complete the first diamond system. Observe- in which of the two square systems now open to you the terminal square is located. Give it the go-by. Pass to the other square system ; then to the remaining diamond 1 system, and lastly to the square system in which the terminal is located, taking care that this quarter of the board receives your last attention :

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But let the terminal square be located' in the same or allied system as the starting square : how shall we proceed ? As follows : (1) In the case of the same system. Suppose starting point = Black's Q 3 ; terminal = White's K Kn 4. These are both in the- same square system. Make one leap in thi square system. (Some authorities recom- mend two or more. Dr. Roget recommends- a larger number.) The object is to get at once on to a diamond system, so as to throw this first square system to the end of the process. Then proceed : diamond, square,, diamond, square :