A392305 Compensation points to be given to the presenter such that the game "Fill your basket" with n coins is fair.

1, 1, 1, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6

Offset

1,4

Comments

"Fill your basket" is a two-players game. The "presenter" selects one of three baskets and the "placer" puts one of n coins with values from 1 to n in that basket. This is repeated n times. At the end the placer is awarded the largest sum of the three baskets and the presenter is awarded the second largest sum. Since the placer has an advantage a compensation, to be agreed upon before the game, is paid to the presenter. The goal of both players is to achieve more points than the opponent. a(n) is the compensation which needs to be paid such that the game will end in a draw if both players apply the optimal strategy.

Links

Table of n, a(n) for n=1..16.

Ruediger Jehn, Online version of the game "Fill your basket"

Example

a(3)=1. The placer must not put coin 3 in the first basket, because the presenter would present in the next two turns the second basket and would achieve a 3:3. If the placer puts 1 in the first basket, they either win 4:2 if the first basket is presented again or 3:2 if a different basket is presented where coin 2 must be placed. If the placer puts 2 in the first basket, they either win 5:1 if the first basket is presented again or 3:2 if a different basket is presented where coin 1 must be placed. Hence when keeping coin 3 to the very end, the presenter can always force a result of 3:2 and therefore a(3)=1 coin must be given as compensation to the presenter.

Crossrefs

Cf. A391537, A391538, A391791, A391792.

Sequence in context: A068953 A362960 A189635 * A109785 A058224 A131808

Adjacent sequences: A392302 A392303 A392304 * A392306 A392307 A392308

Keyword

nonn,more

Author

Ruediger Jehn, Jan 06 2026

Status

approved