another counting problem

[cyclops]

I'll continue until you get bored. Don't forget your daily tsume go.
From the same book ( problem 22 ):
Last year I played 600 games. At least one a day. Must there have been a period of consecutive days when I played exactly 129 games?

Discussion: I remember I solved a ( remotely? ) similar problem in GoDiscussions - with lower numbers -, but I have no idea how. Only that it was cumbersome. And that it took a while before the solution was accepted by our ancestors.

[daal]

Again, there seems to be something wrong in the wording of the problem. If there is no restriction, of course you could have played all 600 games in one day (at 2mins 24 secs pro game ) and there would be no period of consecutive days.

[lorill]

Again, there seems to be something wrong in the wording of the problem. If there is no restriction, of course you could have played all 600 games in one day (at 2mins 24 secs pro game ) and there would be no period of consecutive days.

"At least one a day."

[daal]

Again, there seems to be something wrong with my reading of the problem. If there is no restriction, of course you could have played all 600 games in one day (at 2mins 24 secs pro game ) and there would be no period of consecutive days.

"At least one a day."

Fixed, thanks.

[cyclops]

Lets try myself to propose a tentative approach. I admit there remains lots to be proven.
I call the 129_condition the condition that every day at least one game is played and there is no period of consecutive days during which exactly 129 games are played.
A basic unit I call a period of 129 days consisting of 128 days with exactly 1 game terminated by one day with 130 games.
Repeating this basic unit many times you get a long series that obeys the 129_condition trivially. The average number of games played each days turns out to be 2 in this case. I propose it is impossible to have a lower average for long series obeying the 129_condition.
The average in the problem is 600/365 = 1,64383562 games per day.
For shorter series as in the problem we can lower the average by appending up to 128 ones after the last basic unit.
In the problem we get the lowest average if we have 2 basic units followed by 107 ones. But this gives us 627 games. So I believe that if the number of games in a year is less than 627 the 129_condition cannot be met.