I am participating (in Europe) in a game where one question is concerning mathematics.

Unfortunately, I am very weak in that domain and I am posting here the problem hoping some member “ll help me.

Note for the webmaster : it is authorized in the regalement of the game to ask anyone for resolving the riddles.

Every morning I put 7,5 (seven and a half) sugars in my coffee.

In my kitchen I have 666 boxes of sugar numbered 1 to 666.

Every morning I roll a dice which has 666 sides also numbered 1 to 666.

For taking the sugars, I take them from the box with the number indicated by the dice.

- if that box contains only entire sugars (not cut), I take 7 entire and I cut one into 2 pieces

- If that box contains already an half sugar, I take it + 7 entire

After having taken my 7,5 sugars, I refill the box with entire sugars. If there is already an half sugar in the box, I fill with 8 sugars and I put the half one on the box.

Example:

On the first day, let’s suppose a box containing 1000 entire sugars and let’s suppose that the dice shows the number of the box:

I take 7 entire sugars, I cut 1 in 2 pieces and I take 1 of them. I place the remaining one on the top of the box.

I have a box containing 992 entire sugars and a half one on the top.

I refill the box with 8 entire sugars and I have a box containing 1000 entire sugar + ½ on the top of the box.

Next time that I “ll visit the box (i.e. next time the dice with 666 sides shall indicate the number of that box) I take the ½ sugar on the top and 7 entire inside.

I have a box containing 993 entire sugars,

I refill the box with 7 entire sugars and I have a box containing 1000 entire sugar like the first day.

A.S.O.

We can conclude that

- At the end of the first day, there is on a box in my kitchen ½ sugar. This is for sure!

- At the end of the second day, we can have till 2 half sugar ( but we can also have 0 if the dice indicates again the same number of box !)

- At the end of the third day, we can have till 3 half sugar. ( but we can also have 0, 1 or 3)

- A.S.O.

The question is:

What is the probability for having in my kitchen EXACTLY 333 half sugar after 123 456 789 days?

Thank you for your help.

Kaji

N.B. I beg your pardon for the mistakes in English but it’s not my mother language.