Will's Stats Emporium

Boad Game Design: A Better Blade

William Medwid — Tue, 30 Jun 2026 07:00:00 GMT

Sharing some progress on the board game I’m designing! This game has been almost 7 years in the making, with my first recorded iterations starting in 2019. In the rules doc. Alongside the rules, I’ve recorded the evolution of every change to the rules, and explanations for the intended effect of each.

The result of all this iteration is, in my opinion, a clever tactical game in which 2 to 4 players will continually outmatch the others by deploying armies that exploit the others’ weaknesses. Each player is trying to conquer four out of the many castles on the board, and do everything it takes to stop the others from doing the same.

This game is played on a modular hexagonal board made by the players, with a unique geometry each game.

The core of the game is the weapon triangle: Spears beats Swords, Swords beats Axes, Axes beats Spears. In order to destroy an opponent’s units, you’ll need to assemble a group larger than theirs, of the type of weapon that beats theirs (with an attacking group of 4+ being counted as larger).

For example, in the scenario below, Purple has two swords next to Red’s one axe. 2 > 1 and swords defeat axes, so purple can destroy red’s axe. And Blue has four axes next to Yellow’s 4 spears. Since axes defeat spears, and a group of 4 attackers is always considered larger, Blue can destroy Yellow’s spears one by one.

This weapons triangle makes for a constantly shifting landscape where your soldiers have very decisive victories and losses.

If you’re interested, you can play this game virtually on Tabletop Simulator! This game is still in development, so any feedback is appreciated.

Brave Rats: First Turn Analysis

William Medwid — Mon, 29 Jun 2026 07:00:00 GMT

My Blog and Brave Rats

To kick off my updated blog site, I’m re-visiting some data from my Master’s thesis on the card game Brave Rats. If you’re not familiar with the game (it’s not too widely known), the simplest explanation is that it’s a series of simultaneous-move card reveals, where the player with the highest card will win the round, but each card has a special ability that may change the outcome. First to four victories wins. For reference, the card abilities are as follows:

Strength	Name	Ability
0	Musician	Round is a tie.
1	Princess	Wins the full game if played against Prince (7).
2	Spy	Next round, your opponent must show their card first.
3	Assassin	The lower card wins this round.
4	Ambassador	If you win with this card, you gain an additional round win.
5	Wizard	Nullifies your opponent’s ability.
6	General	Next round, your card’s strength is increased by 2.
7	Prince	You win the round (meaning that this card beats Assassin (3)).

For a complete explanation, I recommend reading the introduction of my thesis or watching this 2-minute video tutorial:

In this article, I’ll look at three main questions:

How does my set of players play the first turn of Brave Rats?
Have their strategies used with knowledge of the optimal MiniMax strategy?
Are there any weaknesses we can exploit with this knowledge?

Using a dataset of 154 recorded games and the MiniMax solution to the game, we’ll see that human players have converged towards the MiniMax solution, but still show slight exploitable imperfections in their first-turn plays.

The MiniMax optimal solution I’m suggesting could be summarized as a perfectly unpredictable player - they perfectly mix up their choices so that no matter what you play, you’ll never get more than a 50% chance to beat it.

Human Strategies Over Time

Unlike in Chess, where one may find success repeating the same opening move every game, in Brave Rats, any such predictable choices will bring a quick defeat, as every card has a weakness. Thus, one must instead try to be unpredictable and select cards according to a distribution that will be hard to beat. As such, I’ll be lumping many first-turn plays from different players together to create an aggregate understanding of how humans play. This means that instead of treating Brave Rats as a mind game about predicting your opponent, we’ll study it in terms of distributions of cards played.

So, we’ll take a look at how often each card has been played. I’ll make one distinction though: the plays from before I calculated and shared the MiniMax optimal strategy, and those after. Have we learned the ways of the machine?

See code

Study 25-35 Hours a Week? The Life of a First-Year Statistics Major

William Medwid — Wed, 26 Aug 2020 07:00:00 GMT

Not all college slogans are the real inspirational type. Cal Poly happens to have a funnily direct one: Study 25-35 hours per week. At first while touring the school, it seems to be everywhere from posters on the wall to pins and magnets given out at orientation. 2 hours per week per unit, outside lecture, sounds like it’d add up to a whole lot. So, I’ve got to ask, does it match up with what students actually do?

As a bit of a spreadsheet nerd, I’ve found myself in a great position to dive into this question and see if I’ve lived up to the slogan myself. Ever since my second year of high school, consistently recording my time spent on academics in a spreadsheet has been an integral part of my work flow. Each assignment is added as a new row once it’s assigned, and I’ll do my best to describe it and estimate how long it’ll take to complete. This makes a dynamic to-do list where I can punch in and out, recording the time in helper columns. Once the assignment is done and I’ve recorded every minute spent on it, it’s filtered out of sight, clearing the list.

Now, let’s take a look at the most important parts of raw data, as that’s often the best way to understand what we’re working with. This includes every assignment and class over my first year of college. Feel free to explore the data with the interactive table below.

# Libraries 
library(tidyverse)
library(stringr)
library(DT)
library(lubridate)
library(modelr)
library(widgetframe)

# Data
# Note: This data is slightly pre-cleaned so only relevant columns and classes are there, and to avoid excess personal information.

all_quarters <- c("Summer 2019", "Fall 2019", "Winter 2020", "Spring 2020", "Summer 2020")
online_quarters <- c("Spring 2020", "Summer 2020")
units <- c(8,14,16,16,8)
#Best to define these here, they'll matter later.

study_time <- read.csv("collegeFirstYear.csv") %>% 
  mutate(Completed = mdy(Completed))%>% 
  mutate(Assignment = ifelse(Title != "", str_trunc(Title, 45), str_c(Type, Tasks_Predicted, Units, sep = " "))) %>% 
  mutate(week_completed = Completed %>% floor_date(unit="week")) %>% 
  mutate(Quarter = factor(Quarter, levels = all_quarters)) %>% 
  mutate(quarter_units = factor(Quarter,
                             levels=all_quarters,
                             labels=units)) %>% 
  rename(Minutes = Min_Real) %>% 
  filter(Class != "GameDevClub")

widget <- DT::datatable(study_time %>% select(Completed,Class,Assignment,Minutes), rownames = FALSE)
widgetframe::frameWidget(widget, width='100%', height = '100%')

Next, we’ll aggregate these tasks and see how much work there was on a weekly basis.

workloads <- study_time %>% 
  group_by(week_completed, Quarter, quarter_units) %>% 
  filter("Class" %in% Type | Quarter %in% online_quarters) %>% #Only counting weeks where there were classes or the quarter was online.
  summarize(week_workload = sum(Minutes, na.rm=TRUE)/60) %>% 
  ungroup() %>% 
  distinct(week_completed, .keep_all = TRUE)

workloads %>% 
  ggplot(aes(x=week_completed, y=week_workload, color=Quarter)) +
  geom_point() +
  geom_smooth() +
  xlab("Week") +
  ylab("Hours of Academics Completed") +
  ggtitle("Academic workload over time (including lectures)")

#Mean is the statistic of choice because large projects can take a long time and end up with work from one week being counted in another.
conf_interval <- t.test(workloads$week_workload, mu=25, alternative = "less")

class_time_total <- study_time %>% 
  filter(Type=="Class") %>% pull(Minutes) %>% sum(na.rm=TRUE)
study_time_total <- study_time %>% 
  filter(Type!="Class") %>% pull(Minutes) %>% sum(na.rm=TRUE)
percent_in_class <- (class_time_total / (class_time_total + study_time_total))*100 %>% round(digits=1)

Alas, I’ve only spent a mean of 19.7 hours per week, a good ways from 25, let alone 35. And that’s with classes included, which are 29% of that time. Apart from the intensive 4-week 8-unit 2019 summer session, there’s very little case to be made that I ended up having a typical 25-35 hour week, other than to consider the time spent traveling between classes, which, though quite time consuming, which is not included in my data, but could hardly be called studying.

conf_interval_per_unit <- t.test(workloads$week_workload/units, mu=2, alternative = "less")

Another more charitable way to reconcile my schedule with the ideal would be to look at the per-unit time spent. It turns out that’s a more positive 1.7 hours, or the equivalent of 27.7 hours for a 16-unit schedule. This is still including classes but feels almost like victory nonetheless.

Next, let’s see which classes were the ones that never required super rigorous homework?

workloads_by_class <- study_time %>% 
  filter(Class != "General") %>% # Removing non class-specific academic imperatives
  group_by(week_completed) %>% 
  filter("Class" %in% Type | Quarter %in% online_quarters) %>% #Only counting weeks where there were classes or the quarter was online.
  ungroup() %>% 
  filter(Type != "Class") %>% 
  group_by(Class,week_completed, Quarter, quarter_units) %>% 
  summarize(week_workload = sum(Minutes, na.rm=TRUE)/60) %>% 
  ungroup() %>% 
  distinct(week_completed, Class, .keep_all = TRUE)

workloads_by_class %>% 
  ggplot(aes(y=Class,x=week_workload)) +
  geom_boxplot(fill="orangered") +
  xlab("Weekly Workload (Excluding Classes)") +
  facet_grid(rows= vars(Quarter), scales = "free_y") +
  geom_vline(xintercept =2, color="darkgreen", linetype = "dashed")+
  scale_x_continuous(breaks=c(0,1,2,5,10,15))

The chart shows only three classes below the two hour line, which match my memories of the less rigorous yet still valuable courses. Note that of the outlier groups may be exaggerated because a project done slowly over many days only counts on the week when it’s completed, unless I’ve broken the project into multiple component tasks.

So, does Cal Poly require a full 25-35 hours a week of homework? Not for me at least. But I suppose the slogan “study 10-20 hours a week, whatever you’re feeling” would be far less inspiring. Also, I’m fairly certain architecture majors exceed 35 hours by at least a factor of two anyway.