Little explorations with HP calculators (no Prime)

[ijabbott]

(12-26-2018 09:34 PM)pier4r Wrote:  The problem: there are 100 blue Christmas balls and 40 white Christmas balls in a line, randomly ordered. There are Anna and Berta that needs balls for their Christmas trees.

They decide that Anna will pick a continuous chunk of the line of length 70, for example between the position 3 and the position 72, the rest will be picked by Berta.

Could they find a way to ensure that Anna and Berta pick both 50 blue balls and 20 white balls? If yes, is a special arrangement of the balls needed to achieve it?

There will always be a contiguous segment of 70 balls containing exactly 50 blue balls and 20 white balls. If there is a starting position that produces more than 50 blue balls, then to even things out there must be some other starting position that produces fewer than 50 blue balls. Since shifting the starting position by one place changes the number of blue balls by at most 1, there must be some starting position between those two positions that produces exactly 50 blue balls.

(This uses a discrete variant of the intermediate value theorem.)