I would like to know if you can help me in the solution for the next AMC exercise.
The problem name is Boxes and Stones
Paul and Carole like to play a game with S stones and B boxes numbered from 1 to B. Before
beginning the game they arbitrarily distribute the S stones among the boxes from 1 to B − 1, leaving
box B empty. The game then proceeds by rounds. At each round, first Paul chooses a subset P of
the stones that are in the boxes; he may choose as many stones as he wants from as many boxes as he
wants, or he may choose no stones at all, in which case P is empty. Then, Carole decides what to do
next: she can either promote the subset P and discard the remaining stones (that is, those stones not
chosen by Paul in the first step); or she may discard the subset P and promote the remaining stones.
To promote a given subset means to take each stone in this subset and move it to the box with the
next number in sequence, so that if there was a stone in this subset inside box b, it is moved to box
b + 1. To discard a given subset means to remove every stone in this subset from its corresponding
box, so that those stones are not used in the game for the remaining rounds. The figure below shows
an example of the first two rounds of a game.
Paul and Carole play until at least one stone reaches box number B, in which case Paul wins the
game, or until there are no more stones left in the boxes, in which case Carole wins the game. Paul
is a very rational player, but Carole is a worthy rival because she is not only extremely good at this
game, but also quite lucky. We would like to know who is the best player, but before that we must first
understand how the outcome of a game depends on the initial distribution of the stones. In particular,
we would like to know in how many ways the S stones can initially be distributed among the first
B − 1 boxes so that Carole can be certain that she can win the game if she plays optimally, even if
Paul never makes a mistake.
Each test case is described using one line. The line contains two integers S (1 ≤ S ≤ 200) and B
(2 ≤ B ≤ 100), representing respectively the number of stones and the number of boxes in the game.
For each test case output a line with an integer representing the number of ways in which the S stones
may be distributed among the first B − 1 boxes so that Carole is certain that she can win the game.
Because this number can be very large, you are required to output the remainder of dividing it by
109 + 7.
Input sample:2 3
Input sample:8 4
The questions are:
1.If there is any mathematical technique to solve the problem.
2.How to solve the problem in Java language.
Thanks for your attention