**The Four Survivalists**, also known as **the List Problem**, is a math problem devised by Gilbert Martinez in October 2015. It is still being worked on by Martinez.

## Contents

## The Problem[]

**In Numbers**[]

How many ways can you write a list containing the numbers 1, 2, 3, and 4 such that the list satisfies the following conditions:

- At no point does the same number appear consecutively in the list.
- All numbers are shown in the list the same number of times.

Moreover, how many ways are there to write such a list overall?

**In Words**[]

The word version of this problem gives rise to the problem's name.

Zeke, Will, Frank, and Carol are away on a survival trek. Each of them are assigned a certain task every day. The tasks are: cook, scout, guard, and hunter. Here are the following conditions:

- A period (
*p*) is the number of days that pass before each person is assigned a task a certain number (*n*) of times. - No one can hold the same position for two consecutive days until a period is concluded.

Here are the questions:

- How many possible configurations can be developed to assign each person one task, given they can be assigned a task
*n*times? - How many configurations follow the conditions listed above?

**Definitions**[]

Based on the conditions, here's how Martinez defined a period and a configuration.

**Period**[]

Let *n* be the number of times a person is assigned a task.

**Configuration**[]

## Calculations[]

**Let ***n* = 1[]

*n*= 1

Here are all of the configurations for *n* = 1, which we will call *c*_{1}.

ZWFC, ZCFW, ZFCW, ZWCF, ZCWF, ZFWC, WZFC, WCFZ, WFCZ, WZCF, WCZF, WFZC, FZCW, GWCZ, FCWZ, FZWC, FWZC, FCZW, CZFW, CWFZ, CFWZ, CZWF, CWZF, and CFZW.

Thus, |*c*_{1}| = 24.

All 24 configurations follow the conditions.

**Let ***n* = 2[]

*n*= 2

Using *c*_{1}, there are 408 configuration permutations that follow the conditions.

For every configuration in *c*_{1}, there are 17 ways to create a unique permutation. There are 24 configurations in *c*_{1}, therefore 17 * 24 = 408.

There are 24 additional configurations that can be created with the following 12 items:

ZWZW, ZFZF, ZCZC, WZWZ, WFWF, WCWC, FZFZ, FWFW, FCFC, CZCZ, CWCW, and CFCF.

For each of these items, there are 2 other items that can be affixed to it, creating 2 unique permutations. There are 12 items, therefore 12 * 2 = 24.

Adding these permutations together results in 432 configurations, each of which will be grouped into *c*_{2}; |*c*_{2}| = 432.

**Let ***n* = 3[]

*n*= 3

As of now, |*c*_{3}| is unknown.

**Lower Estimate**[]

The lower estimate of the value of *c*_{3} is found with the rate of change between the values of *C*_{2} and *C*_{3}.

The rate of change is then multiplied by the value of *c*_{2} (432), giving it a proportional change equivalent to the change between *C*_{2} and *C*_{3}. This gives us an estimate of the value of *c*_{3}.

The lower estimate of |*c*_{3}| is 3,421,008.

**Upper Estimate**[]

The upper estimate of |*c*_{3}| is found by the quotient of *c*_{2} and *C*_{2}.

The quotient is then multiplied by the value of *C*_{3}.

The upper estimate of |*c*_{3}| is 3,421,440.

3,421,008 ≤ |*c*_{3}| ≤ 3,421,440.

A more precise estimate of |*c*_{3}| can be found by the average of these two numbers.

Therefore, |*c*_{3}| ≈ 3,421,224.

**Let ***n* = 4[]

*n*= 4

As of now, the value of |*c*_{4}| is unknown.

**Lower Estimate**[]

The lower estimate of the value of *c*_{4} is found with the rate of change between the values of *C*_{3} and *C*_{4}.

The rate of change is then multiplied by the lower estimate of value of *c*_{3} (3,421,440), giving it a proportional change equivalent to the change between *C*_{3} and *C*_{4}. This gives us an estimate of |*c*_{4}|.

The lower estimate of the value of *c*_{3} is 112,068,801,072.

**Upper Estimate**[]

The upper estimate of the value *c*_{4} is found by the quotient of *c*_{2} and *C*_{2}. The quotient is then multiplied by the value of *C*_{4}.

The upper estimate of |*c*_{4}| is 112,086,374,400.

112,068,801,072 ≤ |*c*_{4}| ≤ 112,086,374,400.

A more precise estimate of |*c*_{4}| can be found by the average of these two numbers.

Therefore, |*c*_{4}| ≈ 112,077,587,736.

## General Formulas[]

For the total configurations, the general formula is:

Here, *x* is the number of unique items in the list, and *n* is the number of times each item must appear in the list.

As of now, there is no general formula for the exact value of |*c*_{n}|. However, there is a formula for an estimation of the value of |*c*_{n}|:

This formula was devised by Martinez on April 30, 2016. Of course, this formula assumes *x* = 4.

## Additional Questions[]

With this question, Martinez also asks if the values for each of these configurations can be described by a specific function; he also asks what the proportions are between the conditioned configurations and the possible configurations as *n* gets bigger.

Martinez is currently working on finding the answers to these questions.