Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
Take any whole number between 1 and 999, add the squares of the
digits to get a new number. Make some conjectures about what
happens in general.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
For Alison's approach:
What happens to the numbers as you go down the rows?
What happens as you go up the rows?
For Bernard's approach:
Which numbers end in a 0 in row $A_2$?
Which numbers end in a 0 in row $A_3$?
Which of these sequences will hit 1000?
For Charlie's approach:
Can you find a similar method to Charlie's to describe the other rows?
Which descriptions include 1000?