IBM Ponder This #291 · July 2022

The cell doubling game

Let’s assume there are $n$ cells, each containing some amount of balls, in sorted order: $a_1\le a_2\le \dots \le a_n$ where $a_i\in \mathbb{N}$ for $1\le i\le n$.

Our goal is to empty one of the cells, but we can only perform one type of move: Choose two cells, $i,j$ and transfer balls from $a_i$ to $a_j$ until the amount of balls in $a_j$ is doubled.

As an example, assume $k=3$ and we are given the cells $(3,4,8)$. By moving 3 from cell 2 to cell 1, we arrive at $(6,1,8)$. Since we keep the cells in sorted order, we write this as $(1,6,8)$.

A sequence of moves leading to one cell being emptied is

$[(3,4,8),(1,6,8),(2,6,7),(4,4,7),(0,7,8)]$

Your goal: Find a sequence of at most 20 moves emptying one cell from the initial configuration

(855661, 1395050, 1402703, 1575981, 2956165, 4346904, 5516627, 5693538, 6096226, 7359806)

Give the sequence in your solution in the above format.

A Bonus "*" will be given for finding a way to add at most 30,000,000 to the content of the cells given in the previous question, such that there is a sequence of moves emptying all the cells except one. As the solution pass the initial state; there is no need to supply the whole sequence.

Solution

Best opened after a real attempt

To be added.