Weekly Problem 37 - 2011
Rotating a pencil twice about two different points gives surprising results...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Using a ruler, pencil and compasses only, it is possible to
construct a square inside any triangle so that all four vertices
touch the sides of the triangle.
Generalise the sum of a GP by using derivatives to make the
coefficients into powers of the natural numbers.
When is a Fibonacci sequence also a geometric sequence? When the
ratio of successive terms is the golden ratio!
With n people anywhere in a field each shoots a water pistol at the
nearest person. In general who gets wet? What difference does it
make if n is odd or even?
David has given a neat solution and also written a program to calculate big powers in modulus arithmetic.
Go to last month's problems to see more solutions.
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Can you beat Piggy in this simple dice game? Can you figure out
Piggy's strategy, and is there a better one?