The Lineup Puzzle
Solution


Puzzle 1: If you start with a 3 by 3 square that has 8 lines, you can easily modify it to create 10 lines by sliding the four corner points left and right. Now we have 3 horizontal lines, 1 vertical, and the 2 diagonals; we've lost two vertical lines, but we've gained 4 new lines that join the corner points, an endpoint of the middle row, and the middle of the top or bottom row.

Here is a table based on Gruenbaum, giving the number of trees, and the maximum number of rows of 3 that can be produced. Values which are not known to be the best have a question mark.
Number of treesRows of 3
31
41
52
64
76
87
910
1012
1116
1219
1322?
1426?
1531?
1637
1740?
1846?
1952?
2057?


Puzzle 2: you can arrange 10 pennies into 5 rows of 4 by placing them in a star pattern:

 
              *

        *   *   *   *

           *     *
              *
          *       *
      


Puzzle 3: You can arrange 16 points in such a way that there are 15 lines of 4 points [Dudeney]. To do this, place three nested pentagons around a center point. The first pentagon can be of arbitrary size, but symmetric about the center point. To determine the second one, draw a star inside the first pentagon; this star will circumscribe the second pentagon. Draw a star inside the second pentagon to determine the third pentagon. The 16 points then comprise the vertices of the three pentagons, and the center point.


Here is a table based on Gruenbaum, giving the number of trees, and the maximum number of rows of 4 that can be produced. The values are known to be the best possible through N = 12.
Number of treesRows of 4
41
51
61
72
82
93
105
116
127
139?
1410?
1512?
1615?
1715?
1818?
1919?
2021?
2123?

Here is a table based on Gruenbaum, giving the number of trees, and the maximum number of rows of 5 that can be produced. Most of these results have not been proven to be maximal.
Number of treesRows of 5
51
61
71
81
92
102
112
123
133
144
156
166
177
189
1910
2011
3536

Reference:

  1. Pierre Berloquin,
    100 Geometric Games.
  2. Henry Dudeney,
    Amusements in Mathematics,
    Originally published in 1917.
  3. Martin Gardner,
    Tree-Plant Problems,
    Time Travel and Other Mathematical Bewilderments,
    Freeman, 1988.
  4. Branko Gruenbaum,
    New Views on Some Old Questions of Combinatorial Geometry,
    Teorie Combinatorie,
    Volume 1, 1976, pages 451-468.

Back to the The Lineup Puzzle.


Last revised on 17 June 2001.