This month’s problem is based on Problem 6 part (b) of the 2023 Euclid Contest. Here is a modified and rephrased version of that problem:
A square is drawn in the plane with vertices at , and . Two blue lines are drawn with
slope 3, one passing through
and the other through . Two red lines are drawn
with slope , one
passing through and the other
through . What is
the area of the square bounded by the red and blue lines?
The answer to this question is . We suggest convincing
yourself of this before attempting the rest of the problem. The fact
that this small square has an area exactly one tenth of the larger
square suggests that there is a way to answer this question by showing
that exactly squares the size of
the small one should fit into the large square. Let’s explore!
Here is some terminology that we will use in this problem.
Definition: A lattice point is a point
for which and are both integers.
Definition: A square whose vertices are at lattice points is called a
unit lattice square. We will denote by the unit lattice square with vertices
, , , and .
Definition: denotes the line segment
connecting to the point . Note that this line has slope
.
Definition: An – lattice line is a line with slope
that passes through at least one
lattice point.
The next two definitions are more complicated. There are examples
given after they are stated.
Definition: The tricky unit square, , is a modified version of
(see above) with the property
that if a line reaches an edge of , it "jumps" to the opposite
side and continues with the same slope. For example, if a line reaches
the top edge of , it
continues with the same slope from the bottom edge directly below where
it reached the top edge. If a line reaches a vertex of , then it has simultaneously
reached two edges. In this situation, it continues with the same slope
from the opposite vertex of .
Definition: Let be a rational number written
in lowest terms. The – loop on is the line passing through
with slope .
Below are diagrams, from left to right, of the – loop, the – loop, and the – loop on . In each diagram, equal
letters mark places where the line jumps from one side of the square to
the opposite side.
Each of the loops above eventually come back to their starting point
and repeat. This happens because is rational (think about
this!). Notice that in the image of the – loop (the middle image),
the loop starts at instead of
. This is because a line of
negative slope starting at
(on the bottom edge) immediately jumps to the top edge and continues
from . It is worth thinking
about how all four vertices of really represent the same
point.
Below is an image of
with the – loop in blue
and the – loop in red.
These two loops divide
into smaller squares. The squares numbered , , , , , and are split in two pieces each across
edges of . Notice that
both the loops pass thorough the point since the four vertices of the
square are the same point. This means, if we count the four vertices as
one intersection point, the two loops intersect exactly times (count them!). Think about how
this compares to the Euclid problem mentioned earlier.
A square is divided by six lines: three parallel blue lines with slope and three parallel red lines with slope .
The leftmost blue line goes from the bottom-left vertex of the square to a point one-third of the way along the top side of the square. The next two blue lines start at points one-third and two-thirds along the bottom side of the square.
The topmost red line goes from the top-left vertex of the square to a point one-third of the way down the right side of the square. The next red lines start at points one-third and two-thirds down the right side of the square.
These intersecting lines create sixteen regions that combine to form ten identical smaller squares within the square. Viewing these regions as forming a tilted and non-uniform four by four grid within in the square, the regions in the top row, from left to right, are numbered 1, 2, 3, 4, those in the second row are numbered 4, 7, 8, 5, those in the third row are numbered 5, 9, 10, 6, and those in the bottom row are numbered 6, 1, 2, 3. The regions at the centre, 7, 8, 9, 10, are intact tilted squares. The other regions come in pairs consisting of a smaller piece and a larger piece of a square, both with the same number.
For each pair of loops below, draw with that pair of loops, count
the number of times the loops intersect, and count the number of squares
into which the loops divide .
The – loop and
the – loop.
The – loop and
the – loop.
The – loop and
the – loop.
The – loop and
the – loop.
In the remaining problems, and
are positive integers that have
no positive divisors in common other than .
How many line segments make up the – loop?
How many – lattice lines intersect
?
Explain why the number of points of intersection of the – loop and the – loop on is equal to the number of
small squares into which the loops divide . Remember that the four
vertices of represent
the same point.
How many – lattice lines intersect
?
Compute the area of the small squares in created by the – loop and the – loop. Our solution will
take for granted that these small squares all have the same area, but
you might like to think about how to prove this.