Ithaca College

Provost Melanie Stein – opening remarks
Awards Ceremony
President La Jerne Terry Cornish – closing remarks
Refreshments on Klingenstein Terrac

Stein, Melanie

Cornish, LaJerne

Closing Ceremonies (Monday)

Robinson, Eli

Eli Robinson
3/11/2022
Gravitational Field Simulator
Abstract:
Gravitational fields are a challenging topic for any novice physicist. In response to these
difficulties and accompanying common misconceptions, I have developed an interactive
simulation to aid in students' understanding and visualization of these abstract concepts. As such,
the topic is often not explored in full detail until a student is enrolled in upper-level physics
courses, like Classical Mechanics. There are, however, compelling reasons to consider presenting
the topic to students in introductory physics courses. Gravitational fields and electric fields
behave analogously. Because students interact more directly with gravitational fields on a regular
basis, research indicates that an introduction in gravitational fields would benefit students’
understanding of fields in general.
This project aims to help students learn the impacts of gravitational forces and fields in a visual
and interactive way. In order to complete this goal, a gravitational field simulator has been
created which allows students and users to visually manipulate a configuration of masses and
their velocities, and observe the resulting motion. These situations are often too challenging to
tackle analytically, and thus a computational simulator is a valuable tool to students and
instructors alike.
I am creating a simulation that will allow students to visually explore interactions between a
variety of preset mass configurations. The simulation is novel in that it additionally allows
students to create their own mass configurations, and explore their own unique hypothetical
experiments.
The project began as an electric field simulator developed by myself and our research
team.This simulation has been transformed and adapted to apply to the learning objectives
associated with gravitational fields due to the similarities of the two types of fields. The major
difference in the simulation is that the masses in the simulation are able to move due to the field.
This causes the gravitational field to change over time and has benefits and restrictions compared
to the initial electric field simulator. At first the project was built using React Native, a
programming language for mobile applications. The simulation however, was relied too heavily
on React Native's modest visualization engine, and became too computationally expensive for
this language to handle. In response, we transitioned to building this app using a more common
language of JavaScript. The simulator is mostly programmed in a JavaScript package called p5
which is particularly useful for visualization purposes,  and easier for many devices to handle
when displaying onto the browser. In addition to JavaScript the simulator is built using  is a small
amount of HTML and CSS in order to stylize the simulator and create interactable elements.
The gravitational orbits resulting from the simulation's presets are tested against
analytical results both quantitatively (through a series of error measurements) and qualitatively
(with a series of resulting graphs). Relevant situations with the same initial conditions were
modeled both in the simulation presets and analytically on Microsoft Excel with equations. Then
the data were graphed for the purposes of comparisons of orbits. I created a script that allows us
to find the average and maximum distance between the two models in order to find errors
resulting from numerical approximations. This testing will ensure that the simulation is
physically accurate in all cases so that when students are working with the simulation, students
can focus their learning on gravitational fields.
In the future, the simulation is to be used in a classroom in order to help students
understand and grapple with gravitational fields. In order to meet this goal we created learning
goals, which are things that the students should be able to understand, or be able to do after using
the simulation. Based on the learning objectives, features within the simulation were created in
order to help students achieve that goal.


English

Gravitational Field Simulator

McGee, Andrew

Gravitational Radiation from a Binary Black Hole Merger

Kemp-Neilson, Martha

Martha Kemp-Neilson 
Abstract 
 
How do people misperceptions and misconceptions of the effects of cigarette usage effect 
their behaviors and decision making of smoking? 
 
Tobacco is responsible for 7 million deaths every year and leading cause of preventable disease, 
disability, and death globally (GBD 2019). This is also the case for United States. In 2019, 
tobacco is estimated to be responsible for 549 thousand deaths in US (IHME database- last 
accessed on February 10, 2022). Furthermore, based on data on 2019, about 34 million US adults 
smoke cigarettes and over 16 million people live with at least one disease caused by smoking, 
and 58 million nonsmoking Americans are exposed to secondhand smoke (CDC/OSH, last 
accessed in February 10, 2019; and US Health and Human Services 2014).  
 
Tobacco market has another nicotine product called electronic cigarettes and other electronic 
nicotine delivery systems (‘ENDS’) which were imported into the US in 2006 (CASAA 2020).  
In the last ten years, electronic cigarette usage has spiked among young adults.  In 2019, 32.9% 
of youth used an ENDS over the past 30 days, while 10.7% used ENDS frequently; that is, on 20 
or more of the past 30 days (Centers for Disease Control and Prevention 2020). Both academics 
and those in the medical community are studying how nicotine affects this demographic and 
whether electronic cigarettes are a gateway drug for combustible cigarettes.  
Electronic cigarettes divided tobacco control community worldwide, as one group claims 
electronic cigarettes (e-cigarettes) to be a breakthrough invention for tobacco users ceasing their 
addiction to conventional cigarettes. Other group claims that tobacco control are at risk of 
reversing years of public health initiatives which helped to reduce tobacco use and are now 
advancing towards a new addiction with currently unknown long-term health hazards.  
  
E-cigarettes take many different forms, vape pens, vaporizers, e-hookahs are all classified as e-
cigarettes. They can also be referred to as electronic nicotine delivery systems (ENDS) and 
alternative nicotine delivery systems (ANDS). All e-cigarettes have a battery and a heating 
element which turns the e-liquid into a smokable vapor. Some e-cigarettes contain nicotine 
which is derived from tobacco, but they do not contain tobacco. There has not been enough 
research done to show that e-cigarettes are an effective way to stop smoking, and currently US 
FDA has been examining e-cigarettes, so no decisions have been made yet.   
  
The marketing and advertising of these devices has been extensive as companies both promoted 
the products as a healthy alternative to traditional cigarettes and teens are attracted to flavors and 
packaging mirroring popular candies. The manufacturers soon found the product commercially 
lucrative. Yet the public still holds many misconceptions when it comes to e-cigarette usage and 
their health effects.  
 
In this research, my objective is to assess; (1) which demographic and socioeconomic 
characteristics of e-cigarettes users are associated with vaping behavior and (2) how their 
knowledge and perceptions about combustible and e-cigarettes influences their vaping behavior. 
For that I plan to use Global State of Tobacco Poll survey which is funded by the Foundation for 
Smoke Free Foundation (FSFW) and collected by Nielsen in May 2019 in 50 States of US.   
 
I will first examine the binary relationship and then use logistic regression by controlling those 
cofounding factors together. For that dependable variable will be set 1 for those who use e-
cigarettes, and 0 for others. For this analysis I will use R-Studio programming language. The 
data includes only nicotine product users age 18 and 69 years of age. Nicotine products include 
factory made cigarettes, roll-your-own tobacco (RYO), electronic cigarettes (e-cigs) with and 
without nicotine and nicotine replacement products (NRTs- patches and gums). US data has 
2029 observation where 926 are female and 1096 are male.  
 
Our preliminary descriptive analysis (Figure 1) suggests that e-cigs usage decreases by age as 
cigarette usage increases. 
Figure 1: Combustible and electronic cigarette usage by age in US, 2019 
 
Furthermore, in the data 46% of respondents said they use e-cigs, 68% uses combustible 
cigarettes and 27% are dual users. Among e-cig users, majority 68% uses e-cigs with nicotine as 
illustrated in Figure 2. 
Figure 2: Combustible, E-Cig and Dual users, US, 2019.  
 
50%
69% 67%
73%
78%
68%
61%
54%
42%
30%
20%
46%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
AGE18_24 AGE25_39 AGE40_54 AGE55_64 AGE65+ Total
Cigarette & E-Cigarette Users by Age Groups in 
US 2019
%cig user % ecig user
Linear (%cig user) Linear (% ecig user)
 
 
 
References 
 
§ GBD 2019 Diseases and Injuries Collaborators. Global burden of 369 diseases and 
injuries in 204 countries and territories, 1990-2019: a systematic analysis for the Global 
Burden of Disease Study 2019. Lancet. 2020;396(10258):1204-1222. doi:10.1016/S0140-
6736(20)30925-9 
§ US Department of Health and Human Services. The Health Consequences of Smoking-50 
Years of Progress, Atlanta, GA: US Department of Health and Human Services, Centers 
for Disease Control and Prevention, National Center for Chronic Disease Prevention and 
Health Promotion, Office of Smoking and Health; 2014.  
§ Global State of Smoking Poll, 2019. Global State of Smoking Poll 2019 - Foundation for 
a Smoke-Free World (smokefreeworld.org). 
 
 


How Do People Misperceptions And Misconceptions Of The Effects Of Cigarette Usage Effect Their Behaviors And Decision Making Of Smoking?

Bromberg, Daia

Whalen Abstract
Daia Bromberg
Faculty Sponsors: Megan Martinez and Osman Yurekli
Cube Compositions and Polyhedra Blooming
Polyhedra unfolding involves taking a shape made up of polygon faces and pulling it
apart until it is two dimensional, similar to how one might flatten a takeout box to use as
a plate. In cases where this can occur with all the faces intact and without causing
overlaps in the flattened shape, we call the process blooming and the 2D product a net.
For this project all the polyhedra are cube compositions: shapes that could be created
starting from just a single cube, then adhering additional cubes face-to-face.
A single cube, for example, has eleven different unfolding options, and none of them
produce overlaps. Adding another cube keeps the same number of faces and nets, but a
third one protruding from the side, as shown on the left in the figure above, will increase
the total faces by 2. On the right, one possible option for a net of this cube composition
is displayed. This shape also has the potential for overlapped unfoldings that are not
valid nets. This generates questions concerning which cube compositions can bloom
into a net, or which nets can be folded up to produce valid polyhedra.
Within these broad ideas, the formation of such shapes can be explored and their
properties tested. This project explores examples of cube compositions that don’t bloom
(their unfoldings always have overlaps), the existence of cube compositions with any
number of faces, and the changes that take place in the amount of faces when a single
cube is added to an existing composition.


Cube Compositions and Polyhedra Blooming

Mitchell, David

Hammel, Annalee

The impact of three reduced exertion exercise prescriptions on 
VO2max  and vertical jump 
Annalee Hammel, David Mitchell 
Ithaca College Department of Exercise and Sport Sciences  
Purpose: Research by Metcalf et. al (2016) shows that reduced exertion high intensity interval 
training (REHIIT) significantly improves cardiorespiratory fitness (CRF) by 12 %. REHIIT 
involves 2 to 3, 20 s cycling sprints against 5% body mass (BM), 3 d.wk-1 for 4 to 6 wk and is 
much shorter in duration and lower in intensity than many traditional exercise programs. Our 
purpose is to determine if lower intensity cycle sprints improve CRF and anerobic power using 
three various protocols. Methods: 18 subjects (21.3 ± 0.5 y, 77.4 ± 23.3 kg, 38.6 ± 10.5 
ml/kg/min) were matched paired based on VO2max into one of three groups, which completed 
three training sessions per week for four weeks. The control group performed 2, 20 s sprints at 
5% BM, whereas the experimental groups completed 2, 20 s sprints at 4% BM or 2, 15 s sprints 
at 5% BM. Sprints were separated by 160 seconds of active rest. Physical self-efficacy, blood 
pressure, vertical jump height, and CRF were measured before and after training. Paired sample 
and independent t-tests will be used to compare within group and between group data, 
respectively; a will be set at 0.05. 
 
Introduction: Over the last 30 years obesity and its complications have become a leading killer in 
America. As of 2017 the CDC estimated that 42.4% of the population in America is obese. There are a 
multitude of factors that can contribute to the prevalence of obesity in an individual, such as sleep habits, 
medications, genetics, and social/economic class standing (NIH). Obesity is commonly diagnosed when a 
person has a BMI over 30. Many complications can arise as a result of obesity such as cardiovascular 
disease, type II diabetes, stroke, and certain types of cancer, all which can lead to poor quality of life and 
an increased risk of death. To combat these complications the United States has increased its spending on 
obesity related conditions (CDC). While exercise is an effective tool to combat obesity, many Americans 
do not meet the recommended guidelines set by the American Heart Association (AHA) and American 
College of Sports Medicine (ACSM). These guidelines recommend 30 minutes of moderate-intensity 
aerobic activity at least five days per week or 20 minutes of vigorous activity at least three days per week. 
The main purpose of this exercise is to improve cardiorespiratory fitness (CRF) which is indicated by an 
increase in VO2 max.  
Many individuals who cannot meet the recommendations set by the AHA and ACSM often cite 
lack of time as the cause. As a result, researchers look for more time efficient ways to increase VO2 max. 
Two of the most common approaches previously utilized to improve VO2 max are high intensity interval 
training (HIIT) and sprint interval training (SIT). HIIT consists of multiple (4-10 repetitions) brief bouts 
(20s-5 min) of high intensity exercise (80%-100% peak heart rate) broken up with rest or low intensity 
workloads (Cuddy 2019).  SIT consists of repeated (6-10 bouts) 20-30 second all out supramaximal 
sprints (>100% VO2max) (Cuddy 2019). Data shows that these training programs are as effective as 
moderate intensity continuous training (MICT) at increasing cardiovascular fitness. (Gillen 2014, Cuddy 
2019) Further, data show that individuals enjoy interval training (IT) to a similar degree as MICT, and in 
some cases even prefer it (Heisz 2015, Jung 2014). While these programs offer solutions to the time 
constraint problems there are concerns about the effectiveness and safety of participants who are more at 
risk of cardiovascular disease. Current protocols have been developed based on data collected from 
younger and more fit individuals which poses an uncertainty around those who are sedentary, overweight, 
and middle-aged. On top of this, older populations may not be willing or capable to complete HIIT/ SIT 
sessions. Consequently, a newer program has emerged known as reduced-exertion high-intensity interval 
training (REHIT). 
REHIT is a relatively new concept that involves one to two “all out” intervals against a 5% 
bodyweight resistance. Studies show that REHIT improves CRF as effectively as HIIT and SIT (Metcalfe 
2016, Metcalfe 2010, Cuddy 2019, Nalcakan 2017, Gillen 2014). While the same benefits are reaped the 
workload is significantly lower, making this a more ideal and accessible mode of exercise. There is still a 
need for more research on the psychological effects of REHIT, however one study found that seven out of 
eight subjects preferred REHIT over HIIT and MICT. 
In this study, we will examine the effects that three bouts of REHIT at varying intensities and lengths will 
have on subjects CRF. Specifically, VO2max will be measured. The three cycle sprint protocols will vary 
in intensity and duration. The control group will complete 20 second intervals at 5% body weight as it has 
been shown to increase VO2max (Metcalfe et al. 2017). Another group will complete 20 second intervals at 
4% bodyweight, and the final group will complete 15 second intervals at 5% bodyweight. Self-efficacy 
will be measured to determine the psychological effects endured by each group. Our goal is to determine 
the protocol that provides the most cardiovascular and psychological benefits. 
Methods 
 
Subjects 
60 undergraduate students (x males and y females) from Ithaca College’s School of Health 
Sciences and Human Performance participated in this project. The subjects were x.x.± SD years 
old, moderately active (x.x ± SD ml.kg.min). Prior to any testing, subjects completed a medical 
history questionnaire to make sure there were no exclusion criteria present, such as current 
orthopedic injuries, allergies, or illnesses that could worsen with exercise. Upon arrival at the lab 
on training days, subjects completed a 24-hour health history form. This form ensured that the 
pre-training and pre-testing instruction compliance was met. Testing/training was rescheduled 
for another day if not. An informed consent was obtained from all subjects for this project, which 
was approved by the Ithaca College IRB. 
Study Design 
 The subjects were divided into three groups with match pairing based on pre-training 
VO2 max to so that there was no initial difference in this key dependent variable. The three 
subjects with the highest VO2max were placed each into a groups A, B, and C. The next three 
highest scores were then placed into groups C, B, and A. This process was repeated for the 
placement of the remaining participants to ensure equal randomization of each group. All 
subjects were required to complete 2-interval workouts 3 days a week for 4 weeks, with intervals 
separated by 160 seconds of active recovery. One group performed 15 second intervals at 5% 
body mass. The second group performed 20-second intervals at 4% body mass. Finally, our 
control group performed 20 second intervals at 5% body mass. This workout structure was 
assigned for the control group because increases VO2 max by 12% across several studies (Gillen 
et al. 2014, Metcalfe 2011).  The following variables were measured before and after training: 
VO2max, vertical jump, waist to hip ratio, and self-efficacy. VO2 was taken as a measure of 
cardiorespiratory fitness, vertical jump as a measure of power, waist to hip ratio as a measure of 
fitness, and self-efficacy as a psychological variable asked after each bout of exercise.  
VO2 Max testing 
We measured VO2 using a graded exercise test on a Monark cycle ergometer (Monark, 
Stockholm, Sweden). Tests were preceded and followed with a warmup and cooldown.  The 
warmup consisted of 5-minute cycling at 80 RPM and a resistance of 1 and 0.5 kg for men and 
women respectively. The cooldown consisted of 5 min of unloaded pedaling at a self-selected 
cadence. The test consisted of 2 min stages at 80 rpm; load was increased by 0.5 kg until 
volitional exhaustion. Expired gas was collected with a metabolic cart (ParvoMedics True One 
2400, Sandy UT), HR with a HRM (Polar: Lake Succcess, NY), and RPE with the traditional 
scale.  The metabolic cart was calibrated before each test. The following criteria were used to 
confirm VO2max was obtained: HR was within 10 bpm of age predicted HRmax, RPE >17, and 
RER was >1.10 
Vertical Jump Testing 
The subjects also completed vertical jump tests prior to, halfway through, and at the end 
of training just twice—pre-post. The jumps were performed in the same manner for each trial, or 
the subject was asked to re-do. For this study the subjects were asked to perform a static jump, 
with their hands on their hips to ensure similar jump technique throughout trials. Subjects 
performed semi squats and held for two seconds prior to jumping. The depth of the semi squat 
was self-chosen by the subject. Three jumps were performed at each testing session, with the 
highest jump being recorded. This variable was measured to assess any changes in lower body 
power indicated by an increase in jump height after interval training. Testing for vertical jumps 
was performed using a timing mat that measured jump height (Just Jump, Probiotics Inc., 
Huntsville, AL). 
Self-Efficacy 
Self-efficacy of physical activity was measured via questionnaire before, two weeks into 
and after training. The physical self-efficacy scale is a reliable tool for the assessment of 
perceived physical confidence. This scale is broken up into two subscales, perceived physical 
ability, and physical self-presentation confidence. The scales were numbered and following 
collection of data scores were totaled and recorded. 
 
 
 


The Impact Of Three Reduced Exertion Exercise Prescriptions On VO2max And Vertical Jump

Turberg, Rachel

TITLE: Using Wearable Technology to Assess Return to Function Following a Total Knee Arthroplasty: 
A Case Report 
PRESENTOR/ AUTHOR: Rachel Turberg 
PRESENTATION TYPE: Oral Presentation for Award 
FACULTY SPONSOR/ MENTOR: Vanessa Mirabito and Kristopher Bosela  
INSTITUTION: Physical Therapy, Ithaca College, Ithaca, NY, United States. 
 
Abstract 
Background and Purpose: Osteoarthritis (OA) is a common pathology affecting adults globally with the 
knee joint being the most affected joint.1 During the late stages of OA, many patients choose to undergo a 
total knee arthroplasty (TKA) to resolve their knee OA symptoms.2 These patients are typically referred to 
physical therapy to address their post-operative symptoms. Walking is part of early rehabilitation 
following TKA and is often a main functional goal for treatment. With the increasing popularity of 
wearable fitness tracking devices, such as smart watches and Fitbits©, there is an opportunity for clinical 
professionals to utilize a patient's personal fitness device to track walking progress. However, there is 
little research into the use of this technology for this purpose. The purpose of this case report is to explore 
the use of personal fitness technology to measure the number of steps per day as an outcome following 
TKA. 
 
Methods and Case Description: The patient was a 61-year-old female referred to outpatient physical 
therapy (PT) services following an elective right TKA. She did not receive pre-surgical rehab for her 
knee. Following surgery and a brief stay in acute care, the patient presented to outpatient physical 
therapy. Her goals included returning to work, returning to golfing, and improving functional mobility to 
enjoy time with family. The primary concerns at the time of examination included increased lethargy, 
pain management, and returning to work. The patient had a good prognosis of returning to her prior level 
of function including independent ambulation and recreational walking. Throughout treatment the patient 
was provided a home exercise program. Walking was highly encouraged as part of her plan of care, but 
no specific goal was set for steps per day. 
 
Outcomes: The patient was seen for 21 visits over 11 weeks. The Knee Outcome Survey was 
administered 3 times during the plan of care. On the 15th visit she scored 71.43, on the 19th visit she 
scored 72.86, and on her final visit she scored 80.00.  Five of seven goals set by the therapist were met 
prior to discharge. The patient’s Fitbit© was used to track daily steps numbers beginning with the day of 
surgery and concluding with discharge. The step values were compared with normative data for the 6-
Minute Walk Test to assess the patient’s progress against standard and normative values. The patient 
improved from 33 steps per day on day of surgery to >8,000 steps per day on date of discharge. The 
difference between these values exceeded the meaningful change of 1,303 steps per day. 
 
Discussion: The use of wearable technology, with proper compliance of wearing the device, may help to 
give therapists insights on how active a patient is truly being. This can be utilized for goal setting related 
to prior level of function if therapist has access to past data and continued tracking throughout 
rehabilitation. In this case, outcomes measured including strength testing, range of motion, KOS, and 
number of steps per day all improved over the course of care. However, the data is insufficient to 
determine how well correlated steps per day are with the other outcome measures. As wearable 
technology continues to evolve and becomes increasingly popular across all patient populations, its use by 
physical therapists may allow for more accurate tracking of patient activity outside of the clinic. Future 
research in physical therapy should be targeted at wearable fitness technologies as it may allow us to 
determine the strength of the relationship between steps per day and other functional outcomes.  
 
References:  
1. Michael JW, Schlüter-Brust KU, Eysel P. The epidemiology, etiology, diagnosis, and treatment 
of osteoarthritis of the knee [published correction appears in Dtsch Arztebl Int. 2010 
Apr;107(16):294]. Dtsch Arztebl Int. 2010;107(9):152-162. doi:10.3238/arztebl.2010.0152 
2. Jette DU, Hunter SJ, Burkett L, et al. Physical Therapist Management of Total Knee 
Arthroplasty. Physical therapy. 2020;100(9):1603-1631. doi:10.1093/ptj/pzaa099 
 
KEYWORDS: Total Knee Arthroplasty, Total Knee Replacement, Fitbit©, Walking 
 
 
 
 
 
 
 


Using Wearable Technology to Assess Return to Function Following a Total Knee Arthroplasty: A Case Report

Scanlon-Black, Ro

“Indicators of Gun Ownership: Findings from the 2018 General Social Survey" 
This study looks at the predictors of having owning a gun. This includes owning a gun, having 
one in your house, or participating in hunting. This is an important study as the Children’s Hospi-
tal of Philadelphia: Center for Violence Prevention state, “According to a 2019 study, gun in-
juries are the second-leading cause of death among U.S. children and teens and the leading cause 
of death of among high school students. 1.7 million children live with unlocked, loaded guns - 1 
out of 3 homes with kids have guns.” The study of indicators of gun ownership has a focus on 
political affiliation and as well as dedication to religion and whether or not that determines or 
correlates with a person having and owning a gun in their home. This study also determines fac-
tors that influence their political affiliation and religious dedication including age, socioeconom-
ic status, and personality. Secondarily I focus on how a person's sex plays a role in whether or 
not they own a gun in addition to their political affiliation and dedication to religion. My hypoth-
esis is that people who are more dedicated to religion and people who are more conservative are 
more likely to own a gun. I am also hypothesizing that women will more likely have the same 
affiliations as their husbands because of their socialization to adhere to their spouses’ beliefs. 
These beliefs or affiliations include political, religious, and opinions on guns. The information is 
found with the General Social Survey of 2018 data set. Preliminary findings show that there is a 
significant relationship between gender, political affiliation, religious beliefs, and gun ownership. 
Specifically, it shows that identifiers such as male, Christian, or conservative have higher rates of 
owning guns or have positive attitudes towards guns. This study shows how political affiliation, 
religion, race, and gender, or social network have a larger effect on gun ownership compared to 
predictors such as personality, which have shown to have little or nothing to do with gun owner-
ship. Using the General Social Survey of 2018 indicators of gun ownership that allude to affilia-
tions have shown more statistical significance with gun ownership than personality. Affiliation 
includes social groups, which often can be grouped in religion, race, gender, networks, or politi-
cal affiliation. Groups that feel more comfortable with each other will often share similar values 
including the value of owning a gun. These networks or social groups also include predictors like 
whether or not the respondent spent time in the armed forces as well as whether or not the re-
spondent owns a gun. Predictors that fall under personality include an individual's level of trust 
with other individuals as well as the level of safety an individual feels in their community. There 
are some indicators that were looked at during the research that does not fall into either of these 
categories. These indicators include satisfaction with life as well as suicide rates in a given area 
compared to the number of guns owned in a specific area. Basic findings show how few indica-
tors that fall under the category of personality or feelings do not have as statically strong a rela-
tion as indicators that refer to demographic and social grouping. 


Indicators of Gun Ownership: Findings from the 2018 General Social Survey

Closing Ceremonies (Tuesday)

Woodworth, Jamie

Title: Deriving the Expected Value of Blank Chutes & Ladders Boards
Name: Jamie Woodworth
Faculty Sponsor: Emilie Wiesner
Chutes & Ladders is a children's board game where the goal is to move a token from the start space to
the end space. A fair spinner determines the number of spaces the token travels for each turn, and the
game ends when the token lands exactly on the end space. If the player spins a value higher than what
is needed to reach the end space, the token remains on the current space. This project focused on
blank boards, which are boards that do not contain any chutes or ladders. The main goal is to find the
average number of turns it takes to win a blank board, which is called the expected value of the board.
On blank boards, the expected value depends only on the size of the board s and the value of the
spinner n. In previous work, the expected value for blank boards using a spinner value of 1 and spinner
values equal to or greater than s were derived analytically [1]. The method used to show those two
cases becomes difficult to solve for cases where 2<n<s. This presentation will focus on deriving the
expected value when n=2.
The expected value is computed as the sum of winning in all possible turns k multiplied by the
probability of winning in k turns. This expression produces an infinite sum since k ranges from 1
towards infinity. This is due to the token having to land exactly on the end space. There are several
lemmas that aid in deriving a closed form expression for the expected value of blank boards.
The first lemma states that the minimum number of turns required to win a board of size s (an
s-board) using a spinner value of n (an n-spinner) is given by s divided by n, rounded up to the nearest
whole number. This lemma establishes the starting value for the sum, since the probability of winning in
a number of turns less than the minimum is zero. A total number of s spaces must be traveled to reach
the end space, so spinning the highest value n on the spinner as many times as needed minimizes the
number of turns.
The second lemma is the derivation of the counting function, which recursively counts the number
of ways to win an s-board using an n-spinner in k turns. This counting function is needed in order to
calculate the probability of winning in k turns. The key insight of this lemma is that if the token travels
one space in the first turn, it now sees itself on a board of size s-1 with k-1 turns left before winning.
This pattern continues for all possible numbers of spaces traveled in the first turn.
The third and fourth lemmas are used to collapse the infinite sum. The third shows that the counting
function becomes a constant value for all k>s when n=2. This allows the counting function to be taken
outside of the infinite sum. For large values of k, winning can only be delayed by repeatedly spinning a
value of 2 each turn to avoid landing on the end space.
The final lemma needed shows that the sum from k=1 to infinity of k divided by 2^k can be written
as a finite sum. This allows for a finite expected value to be calculated. Since the partial sum of k/2^k
converges, taking the limit as k tends towards infinity also shows that the infinite sum converges.
By using all of the above lemmas, the average number of turns it takes to win a game of Chutes &
Ladders using a 2-spinner can be written as a finite value. The expression found matches numerical
predictions made in previous investigations, and is similar to a documented sequence studied in
combinatorics [2]. The method used for the n=2 case could be applied to other cases in the
hard-to-solve range, and a general form could be derived.
Bibliography:
[1] Woodworth, J. (2021). Expected Values of Blank Chutes \& Ladders Boards. Proceedings of the
Research Experience Course, 1(2), 22-29.
[2] Problem 11623. American Mathematical Monthly, Vol. 119, February 2012.
