Designs on square Grids
Welcome to, as usual, three new puzzles. However, unlike previous columns,
where solutions to two were known (and included in the related Solutions and
Sources in the next issue), this time expect to see solutions to all three in June.
1.In composing this puzzle, writer/mathematician Barry Cipra was inspired by a work called “Straight Lines in
Four Directions and All Their Possible Combinations” by the
great American artist Sol Lewitt. You are given 16 unit squares,
each containing a different combination of vertical crossing
line, horizontal crossing line, SW-NE diagonal, and SE-NW
diagonal. The object is to tile a 4× 4 grid with these squares
(without rotating them) in such a way that no line ends before it
hits the edge of the grid or, alternatively, prove it cannot be done.
Figure 1 shows a failed attempt, with dangling ends circled.
2.Cipra also composed an entertaining curved version. In it, each unit square contains one of the 16 possible
combinations of four quarter circles, each of radius 1/2
and centered at a corner. Can you tile a 4× 4 grid with these
squares so no path ends before it hits the edge of the grid?
Or, better still, can you arrange the tiles so an even
number of quarter circles meet at each edge shared by
two squares? Figure 2 shows a failed attempt, again with
dangling ends circled.
3.We now upgrade to a 5× 5 grid in a puzzle passed to me by my Dartmouth colleague Vladimir Chernov.
In it, each unit square is blank or has one diagonal drawn in.
How many non-blanks can you pack in without any two
diagonals meeting at a corner? Figure 3 shows a legal
configuration with 15 diagonals. Can you beat it with 16
diagonals? Or prove 15 is the maximum possible?
Readers are encouraged to submit prospective puzzles for future columns to firstname.lastname@example.org.
Peter Winkler ( email@example.com) is William Morrill Professor of Mathematics and computer Science
at dartmouth college, hanover, nh.