Squares
Coordinates of the centre of the 20th square: (59,21)
Coordinates of the bottom left hand vertex of the 34th square: (100,33)
Coordinates of the centre of the -15th square: (-46,-14)
Triangles
Coordinates of the top vertex of the 23rd triangle: (90,10)
Coordinates of the top left hand vertex of the 58th triangle: (228,5)
And more squares
Coordinates of the centre of 22B: (44,-36)
Coordinates of the left hand vertex of 26G: (48,-46)
We received good solutions from Ivan Ivanov from the 47th High School in Sofia, Bulgaria and from Michael Sena from NSBH. Well done to you both.
Ivan reasoned as follows:
SQUARES
1. If $C(n)$ is the centre of square $n$,
then the coordinates of $C(n)$ satisfy the equations: $x(n) = 3n - 1$, and $y(n) = n + 1$.
2. If $L(n)$ is the bottom left hand vertex of square $n$,
then the coordinates of $L(n)$ satisfy the equations: $x(n) = 3n - 2$, and $y(n) = n - 1$.
Given the above equations:
a. The coordinates of the centre of the $20$th square are $(59,21)$
b. The coordinates of the centre of the $-15$th square are $(-46,-14)$
c. The coordinates of the bottom left hand vertex of the $34$th square are $(100,33)$
TRIANGLES
1. If $C(n)$ is the vertex of triangle $n$,
then the coordinates of $C(n)$ satisfy the equations: $x(n) = 4n - 2$, $y(n) = 10$ when $n$ is odd and $y(n) = 0$ when $n$
$L(n)$ is the left top (bottom) vertex of triangle $n$,
then the coordinates of $L(n)$ satisfy the equations: $x(n) = 4n - 4$, $y(n) = 5$.
Given the above equations:
a. The coordinates of the vertex of the $23$rd triangle are $(90,10)$
b. The coordinates of the left top vertex of the $58$th triangle are $(228,5)$
AND MORE SQUARES
1. If $C(n)B$ is the centre of square $nB$,
then the coordinates of $C(n)B$ satisfy the equations: $x(n)B = 2n + 2$, and $y(n)B = (-2)n + 10$.
2. If $L(n)G$ is the left hand vertex of square $nG$,
then the coordinates of $C(n)G$ satisfy the equations: $x(n)G = 2n - 4$, and $y(n)G = (-2)n + 6$.
Given the above equations:
a. The coordinates of the centre of square $22B$ are $(46,-34)$
b. The coordinates of the left hand vertex of square $26G$ are $(48,-46)$
Michael wrote a short program for each problem that produced the same resultsere is his solution to SQUARES: