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 could have?
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?