We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
Can you prove that the sum of the distances of any point inside a
square from its sides is always equal (half the perimeter)? Can you
prove it to be true for a rectangle or a hexagon?
A moveable screen slides along a mirrored corridor towards a
centrally placed light source. A ray of light from that source is
directed towards a wall of the corridor, which it strikes at 45
degrees. . . .
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Draw a line (considered endless in both directions), put a point
somewhere on each side of the line. Label these points A and B. Use
a geometric construction to locate a point, P, on the line,. . . .
Neil has successfully generalised his results for the Multiplication square.
Go to last month's problems to see more solutions.
An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.
This game is known as Pong hau k'i in China and Ou-moul-ko-no in Korea. Find a friend to play or try the interactive version online.