Hannah tackled the first part of this
problem. Here are the squares she drew:
(a)

(b)

(c)
(d)

(e)
In all of these, the side AB could either
be from the bottom corner to the right, or from the left corner to
the top. Do you see why?
Hannah went on to complete the arrow
notation for these squares. She's taken the side AB to be from the
bottom up to the right.
(a) $A 1 \rightarrow +1 \uparrow B 1 \leftarrow + 1 \uparrow C 1
\leftarrow + 1 \downarrow D 1 \rightarrow + 1 \downarrow A$
(b) $A 2 \rightarrow + 1 \uparrow B 1 \leftarrow + 2 \uparrow C 2
\leftarrow + 1 \downarrow D 1 \rightarrow + 2 \downarrow A$
(c) $A 3 \rightarrow + 1 \uparrow B 1 \leftarrow + 3 \uparrow C 3
\leftarrow + 1 \downarrow D 1 \rightarrow + 3 \downarrow A$
(d) $A 2 \rightarrow + 2 \uparrow B 2 \leftarrow + 2 \uparrow C 2
\leftarrow + 2 \downarrow D 2 \rightarrow + 2 \downarrow A$
(e) $A 3 \rightarrow + 2 \uparrow B 2 \leftarrow + 3 \uparrow C 3
\leftarrow + 2 \downarrow D 2 \rightarrow + 3 \downarrow A$
Good work, Hannah!
Ahmed gave us instructions to construct a
square where you are given one of its sides:
Suppose you're given the side $A a \rightarrow + b \uparrow B$. (I
noticed that $\downarrow$ could be written as $-\uparrow$ and
$\leftarrow$ could be written as $-\rightarrow$, so $a$ and $b$
could be negative, but that doesn't matter. This makes it a bit
easier, as we only have two sorts of arrow then.) Then the square
is either $$A a \rightarrow + b \uparrow B -b \rightarrow +a
\uparrow C -a \rightarrow -b \uparrow D b \rightarrow -a \uparrow
A$$ or $$A a \rightarrow + b \uparrow B b \rightarrow -a \uparrow C
-a \rightarrow - b \uparrow D -b \rightarrow + a \uparrow
A.$$
Well done, Ahmed, especially for spotting
that there are two possible squares.
Ahmed then worked out which of the collections of points could
be a square.
- (8,3), (7,8), (2,7), (3,2). In arrow notation, this would be $A
-1 \rightarrow +5 \uparrow B -5 \rightarrow - 1 \uparrow C 1
\rightarrow - 5 \uparrow D 5 \rightarrow + 1 \uparrow A$. This is
of the first form, with $a=-1$ and $b=5$. So this is a square.
- (3,3), (7,4), (8,8), (4,7). In arrow notation, this would be $A
4 \rightarrow + 1 \uparrow B 1 \rightarrow + 4 \uparrow C -4
\rightarrow - 1 \uparrow D -1 \rightarrow - 4 \uparrow D$. This
isn't of either form, so the points don't form a square.
- (16,19), (18,22), (21,20), (19,17). In arrow notation, this
would be $A 2 \rightarrow + 3 \uparrow B 3 \rightarrow - 2 \uparrow
C -2 \rightarrow - 3 \uparrow D -3 \rightarrow + 2 \uparrow A$.
This is of the second form, with $a=2$ and $b=3$, so the points
form a square.
- (4,20), (21,19), (20,2), (3,3). In arrow notation, this would
be $A 17 \rightarrow - 1 \uparrow B -1 \rightarrow - 17 \uparrow C
-17 \rightarrow + 1\uparrow D 1 \rightarrow + 17\uparrow A$. This
is also of the second form, with $a=17$ and $b=-1$, so the points
form a square.