### Pebbles

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?

### 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.

### 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?

# Maximum Fibonacci

##### Stage: 3 Short Challenge Level:

999

Let the first two terms of the sequence be $a$ and $b$ respectively. Then the next three terms are $a+b$, $a+2b$ and $2a+3b$. So $2a+3b=2004$. For $a$ to be as large as possible, we need $b$ to be as small as possible, consistent with their both being positive integers. If $b=1$ then $2a=2001$, but $a$ is an integer, so $b\neq 1$.

However, if $b=2$ then $2a=1998$, so the maximum possible value of $a$ is $999$.

This problem is taken from the UKMT Mathematical Challenges.
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