Sorry for the last problem- made a typo. If you want another chance, here you go. Good Luck!

Divide the circle into n sectors. In the beginning, each sector has one dot in it. In other words, there are also n dots. 1. Choose one sector.

Take one dot away from that sector, and put it in the next sector, going counter -clockwise.

Take two dots away from the next sector (the sector that you just put a dot in step two), and put it in the next sector, still going counter-clockwise .

Take one dot away from the third sector (the sector that you just put two dots in step 3), and put it in the next sector, still going counter-clockwise.

If a sector is empty, DO NOT put a dot in there. Instead, put the dots you needed to put in the blank sector to the next non- blank sector, still going counter-clockwise.

Keep on doing this UNTIL:

All the dots end up in one sector. When this happens, the number of dots is called HAPPY

The pattern starts repeating (in other words, they never end up in one sector). When this happens, the number of dots is called UNHAPPY.

For example, the number 5 is happy because all 5 dots end up in one sector.

This is how it goes: 1 1 1 1 1

0 2 1 1 1

0 0 3 1 1

0 0 2 2 1

0 0 2 0 3

0 0 3 0 2

0 0 1 0 4

0 0 2 0 3

0 0 0 0 5 which is why 5 is happy. (it is not the answer!)

But, the number 7 is unhappy because the dots never end up in one sector, no matter what.

What is the largest happy number below 50,000,000,000?

For the first part, go here:

https://brilliant.org/profile/jonathan-ydncjm/sets/are-you-happy-or-unhappy/

Challenge: Generalize. Find a formula, and prove it.

(This is a Diamond Middle School problem)

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