### Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

### Modular Knights

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.

### Telescoping Functions

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

# Old Nuts

##### Stage: 5 Challenge Level:

Four people are shipwrecked and there are only coconuts to eat. Being prudent they collect all the coconuts they can find and weary from their work fall asleep.

In the night one of the castaways wakes up and secretly divides the coconuts into four equal piles, she hides her share and throws to the monkeys the three that were left over before putting all the remaining nuts back into one pile.

Later another of the castaways wakes up and she too secretly divides the coconuts into four equal piles, she hides her share and throws to the monkeys the three that were left over before putting all the remaining nuts back into one pile.

Later still, yet another of the castaways wakes up and she too secretly divides the coconuts into four equal piles, hides her share and throws to the monkeys the three that were left over before putting all the remaining nuts back into one pile.

Just before morning the last castaway wakes up and she too secretly divides the coconuts into four equal piles, hides her share and throws to the monkeys the three that were left over before putting all the remaining nuts back into one pile.

Next morning with a much reduced pile the four castaways find they can share out equally all the coconuts that are left!

What is the least number of coconuts they could have started with?

Extension: Solve the generalised problem with n castaways, dividing the pile into n shares, hiding one share and throwing m coconuts to the monkeys each time.