Ten times eighteen

TL;DR

This paper analyzes a probabilistic game involving dice and slotting strategies, presenting an optimal and efficient approach to maximize expected scores, and examining the impact of foreknowledge of rolls.

Contribution

It introduces a polynomial-time strategy for the game, balancing optimality and computational efficiency, and studies the effect of knowing all rolls beforehand.

Findings

Optimal strategy for expected score achieved

Polynomial-time and space complexity strategy developed

Limited score increase with foreknowledge of rolls

Abstract

We consider the following simple game: We are given a table with ten slots indexed one to ten. In each of the ten rounds of the game, three dice are rolled and the numbers are added. We then put this number into any free slot. For each slot, we multiply the slot index with the number in this slot, and add up the products. The goal of the game is to maximize this score. In more detail, we play the game many times, and try to maximize the sum of scores or, equivalently, the expected score. We present a strategy to optimally play this game with respect to the expected score. We then modify our strategy so that we need only polynomial time and space. Finally, we show that knowing all ten rolls in advance, results in a relatively small increase in score. Although the game has a random component and requires a non-trivial strategy to be solved optimally, this strategy needs only polynomial time and space.