Puzzles
9 December
In a 3 by 5 grid of squares, if a line is drawn from the bottom left corner to the top right corner, it will pass through 7 squares:
In a 251 by 272 grid of squares, how many squares will a line drawn from the bottom left corner to the top right corner pass through?
Show answer & extension
Hide answer & extension
The line passes through one square in each column plus an extra square in a column whenever is passes between rows. It must pass between rows 250 times (to get from row 1 to 251) so
in total it passes through 250+272 = 522 squares.
Extension
In a 200 by 300 grid of squares, how many squares will a line drawn from the bottom left corner to the top right corner pass through?
23 December
I draw the parabola \(y=x^2\) and mark points on the parabola at \(x=17\) and \(x=-6\).
I then draw a straight line connecting these two points.
At which value of \(y\) does this line intercept the \(y\)-axis?
2 December
What is the maximum number of lines that can be formed by the intersection
of 30 planes?
Two lines
Let A and B be two straight lines such that the gradient of A is the y-intercept of B and the y-intercept of A is the gradient of B (the gradient and y-intercept of A are not the same). What are the co-ordinates of the point where the lines meet?
Show answer & extension
Hide answer & extension
Let A have the equation \(y = mx + c\). B will have the equation \(y = cx + m\).
Therefore, \(mx + c = cx + m\).
Which rearranges to \(x(m - c) = m - c.\)
So \(x = 1\).
Substituting back in, we find \(y=m+c\).
The co-ordinates of the point of intersection are \((1,m+c)\).
Extension
Let \(a\), \(b\) and \(c\) be three distinct numbers. What can you say about the points of intersection of the parabolas:
$$y = ax^2 + bx + c\mathrm{,}\\
y = bx^2 + cx + a\mathrm{,}\\
\mathrm{and\ }y = cx^2 + ax + b$$
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