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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?

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Happy Numbers

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.

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Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

Odds, Evens and More Evens

Stage: 3 Challenge Level: Challenge Level:1

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?