273.—­THE ROUND TABLE.

Seat the same n persons at a round table on

(n — 1)(n — 2)
--------------
2

occasions so that no person shall ever have the same two neighbours twice.  This is, of course, equivalent to saying that every person must sit once, and once only, between every possible pair.

274.—­THE MOUSE-TRAP PUZZLE.

[Illustration

         6 20 2 19
      13 21
     7 5
    3 18
    17 8
    15 11
     14 16
       1 9
          10 4 12

]

This is a modern version, with a difference, of an old puzzle of the same name.  Number twenty-one cards, 1, 2, 3, etc., up to 21, and place them in a circle in the particular order shown in the illustration.  These cards represent mice.  You start from any card, calling that card “one,” and count, “one, two, three,” etc., in a clockwise direction, and when your count agrees with the number on the card, you have made a “catch,” and you remove the card.  Then start at the next card, calling that “one,” and try again to make another “catch.”  And so on.  Supposing you start at 18, calling that card “one,” your first “catch” will be 19.  Remove 19 and your next “catch” is 10.  Remove 10 and your next “catch” is 1.  Remove the 1, and if you count up to 21 (you must never go beyond), you cannot make another “catch.”  Now, the ideal is to “catch” all the twenty-one mice, but this is not here possible, and if it were it would merely require twenty-one different trials, at the most, to succeed.  But the reader may make any two cards change places before he begins.  Thus, you can change the 6 with the 2, or the 7 with the 11, or any other pair.  This can be done in several ways so as to enable you to “catch” all the twenty-one mice, if you then start at the right place.  You may never pass over a “catch”; you must always remove the card and start afresh.

275.—­THE SIXTEEN SHEEP.

[Illustration: 

+========================+
||    |     |     |     ||
|| 0  |  0  |  0  |  0  ||
+-----+-----+-----+------+
||    |     |     |     ||
|| 0  |  0  |  0  |  0  ||
+========================+
||    ||    |     ||    ||
|| 0  || 0  |  0  || 0  ||
+-----+=====+=====+------+
||    |     ||    |     ||
|| 0  |  0  || 0  |  0  ||
+========================+

]

Here is a new puzzle with matches and counters or coins.  In the illustration the matches represent hurdles and the counters sheep.  The sixteen hurdles on the outside, and the sheep, must be regarded as immovable; the puzzle has to do entirely with the nine hurdles on the inside.  It will be seen that at present these nine hurdles enclose four groups of 8, 3, 3, and 2 sheep.  The farmer requires to readjust some of the hurdles so as to enclose 6, 6, and 4 sheep.  Can you do it by only replacing two hurdles?  When you have succeeded, then try to do it by replacing three hurdles; then four, five, six, and seven in succession.  Of course, the hurdles must be legitimately laid on the dotted lines, and no such tricks are allowed as leaving unconnected ends of hurdles, or two hurdles placed side by side, or merely making hurdles change places.  In fact, the conditions are so simple that any farm labourer will understand it directly.