How often does the Senate Vote In Palindromes?

What are the possible combinations of votes that can result in a palindrome?

When thinking about palindromes, the first thing I need to do is think about what the reversible part of the palindrome is. With senate votes, the first part of the palindrome can be considered the number of “yea” votes while the second part of it can be considered the number of “nay” votes.

Considering the number of seats in the senate, the number of yea votes can be any number between 0 and 100.

Considering the problem of palindromes will narrow down 0:100 in two ways.

First, a unanimous vote in the senate is impossible! Kidding aside, I also know that a unanimous vote can’t result in a palindrome, so the case where there are 100 yea votes can be discarded. In addition, we won’t look at cases were no senators vote!

Second, Looking at votes between 1 and 99 is doubling the amount of work. Votes in the yea column greater than or equal to 50 are palindromes of the number of votes below 50. So it’s only necessary to look at yea votes where the number of votes is between 1 and 49.

For illustrative purposes, let’s put that vector in a data frame

library(dplyr)

votes <- data_frame(yea = 1:49)

## Warning: `data_frame()` is deprecated, use `tibble()`.
## This warning is displayed once per session.

Now that we have a vector of all of the numbers between 0 and 49, we’ll build a vector of their palindromes and put them in a new column for all of the “nay”" votes.

library(stringr)
library(purrr)

votes <- votes %>% 
  mutate(nay = formatC(yea, width = 2, format = "d", flag = "0") %>% 
           str_split("") %>% 
           map(rev) %>% 
           map_chr(paste, collapse = "") %>% 
           as.numeric())

In the above code chunk, I introduced a function it seems a lot of people don’t get to play with very often, formatC, an interface to the c function printf. Basically, it allows users to format a text input. Here, it’s used to add leading zeros.

Above, there’s also a call to str_split from the stringr package. It’s used here to split the two digits so that they can then be reversed.

str_split outputs a list of character vectors where each element is the two characters. To operate on this list, map from the purrr package is used, to operate on each element in the list. In this instance map applies the rev function to reverse each item in each element in the list so that "8" "1" becomes "1" "8".

Similarly, map_chr pastes the items in each element together (changing "1" "8" to "18") and uses the _chr modifier to return a character vector that is then converted into a numeric one.

After computing the nay votes, we need to narrow down our list of yea’s and nay’s to votes that are possible in the 100 seat senate.

votes <- votes %>% 
  mutate(vote_count = yea + nay) %>% 
  # Filter our impossible vote counts
  filter_at(vars(vote_count), all_vars(. <= 100))

If you’ve been playing along at home, you will notice that 10 of the vote counts were dropped.

Now we have all of the possible vote tallies that are palindromic!