In the third case, the smallest number is 100.  The number 1,000 would also do, and there are just seventy-two other numbers between these that the cat might employ with equal success.

233.—­THE ECCENTRIC CHEESEMONGER.

To leave the three piles at the extreme ends of the rows, the cheeses may be moved as follows—­the numbers refer to the cheeses and not to their positions in the row:  7-2, 8-7, 9-8, 10-15, 6-10, 5-6, 14-16, 13-14, 12-13, 3-1, 4-3, 11-4.  This is probably the easiest solution of all to find.  To get three of the piles on cheeses 13, 14, and 15, play thus:  9-4, 10-9, 11-10, 6-14, 5-6, 12-15, 8-12, 7-8, 16-5, 3-13, 2-3, 1-2.  To leave the piles on cheeses 3, 5, 12, and 14, play thus:  8-3, 9-14, 16-12, 1-5, 10-9, 7-10, 11-8, 2-1, 4-16, 13-2, 6-11, 15-4.

234.—­THE EXCHANGE PUZZLE.

Make the following exchanges of pairs:  H-K, H-E, H-C, H-A, I-L, I-F, I-D, K-L, G-J, J-A, F-K, L-E, D-K, E-F, E-D, E-B, B-K.  It will be found that, although the white counters can be moved to their proper places in 11 moves, if we omit all consideration of exchanges, yet the black cannot be so moved in fewer than 17 moves.  So we have to introduce waste moves with the white counters to equal the minimum required by the black.  Thus fewer than 17 moves must be impossible.  Some of the moves are, of course, interchangeable.

235.—­TORPEDO PRACTICE.

[Illustration: 

      10 6 7
       \ |/
    4 u u 2
     \ u /
    3-u u u u
       u u
      u u u u -----9---
     / u
    8 u u
       / \
      1 5

]

If the enemy’s fleet be anchored in the formation shown in the illustration, it will be seen that as many as ten out of the sixteen ships may be blown up by discharging the torpedoes in the order indicated by the numbers and in the directions indicated by the arrows.  As each torpedo in succession passes under three ships and sinks the fourth, strike out each vessel with the pencil as it is sunk.

236.—­THE HAT PUZZLE.

[Illustration: 

      1 2 3 4 5 6 7 8 9 10 11 12
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+
    | *| o| *| O| *| O| *| O| *| O| | |
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+
    | *| | | O| *| O| *| O| *| O| O| *|
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+
    | *| *| O| O| *| O| | | *| O| O| *|
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+
    | *| *| O| | | O| O| *| *| O| O| *|
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+
    | *| *| O| O| O| O| O| *| *| | | *|
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+
    | | | O| O| O| O| O| *| *| *| *| *|
    +—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+—­+

]