Question

Find \(x_{k},\) the number of ways to make a stack of \(k\) poker chips if only red, blue, and gold chips are used and no two gold chips are adjacent. [Hint: Show that \(x_{k+2}=2 x_{k+1}+2 x_{k}\) by considering how many stacks have a red, blue, or gold chip on top.]

Short answer

Use recursion: \(x_{k+2} = 2x_{k+1} + 2x_k\) to find solutions.

Step-by-step solution

Understanding the Problem

We need to find a recursive formula for the number of ways to stack 3 types of chips (red, blue, and gold) where no two gold chips are adjacent. The hint suggests building the stack recursively by adding one chip to a smaller stack.

Base Cases

First, we establish the base cases. For a stack of height 1, we can choose any of the 3 chips, so \(x_1 = 3\). For a stack of height 2, we choose 2 chips with one constraint: no two gold chips can be together. The possibilities are: each chip is different, or both are red or both are blue, so \(x_2 = 8\).

Analyzing the Recurrence

For a stack of height \(k+2\), we consider the chip on top. It could either be red or blue (adding to \(x_{k+1}\) stacks), or it could be gold (which can't be adjacent to another gold, so the stack beneath must end in a non-gold chip from \(x_k\)).

Writing the Recurrence

If the top chip is red or blue, we can take any valid \(k+1\) stack and add a chip: \(2x_{k+1}\). If the top chip is gold, the stack underneath cannot end with a gold chip, so consider valid \(k\) stacks ending with red or blue, prompting another \(2x_k\).

Deriving the Recursive Formula

Thus, the recursive relation as derived from considering all the possibilities is \(x_{k+2} = 2x_{k+1} + 2x_k\). This takes into account all ways to build a \((k+2)\)-stack by leveraging smaller stacks.

Key concepts

Poker Chip Stacking Problem

In the Poker Chip Stacking Problem, we're tasked with finding the number of ways to stack a set of poker chips, specifically red, blue, and gold, such that no two gold chips are adjacent to each other. This intriguing problem is an application of recursive thinking to combinatorial arrangements. When solving such problems, it often helps to start small and build up by increasing the complexity one step at a time. By using a hint that connects stacks of differing heights through a specific recurrence relation, we can systematically build stacks from smaller known configurations.

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. It provides tools for counting combinations, permutations, and various configurations. In problems like the Poker Chip Stacking Problem, combinatorics is key to calculating the number of possible arrangements under specified rules.

• We begin by establishing base cases that are easy to compute manually, such as single-chip and double-chip stacks. Base cases form the grounding for our recursive approach.
• From these base cases, we consider possible combinations of chips, building complex arrangements from simple ones by applying the defined constraints, such as non-adjacency of specific items.
• The recursive formula derived in this context relies heavily on combinatorial reasoning, ensuring all possible legal configurations are counted effectively.

By using combinatorics, students learn not only how to count these configurations but also develop problem-solving strategies that are recursive in nature, enhancing their overall analytical skills.

Non-Adjacent Constraints

Non-adjacent constraints are conditions that dictate how elements in a set can be arranged relative to each other. In our context, the constraint means no two gold chips can be placed next to each other in the stack. Understanding these restrictions is crucial as they direct the permissible configurations that must be considered. When building the stacks:

• Consider the impact of each color's placement, especially gold, given its restriction. If a gold chip is chosen, careful arrangement is required to prevent it from being adjacent to another gold chip.
• The recursive relation factors in these constraints by calculating valid stacks where a gold chip can be placed without violating these rules, which helps in accurately predicting and calculating possible stacks.

To solve problems with such constraints, it's essential to systematically analyze possible positions and ensure that the configuration adheres to these rules, thereby honing the ability to balance creativity and structure in problem-solving.