In the May/June 2026 edition of Contingencies, Josh Feldmen shared three jigsaw-puzzle puzzles. Below, I give the problem statement verbatim followed by my attempt at a solution.
A jigsaw‑puzzle manufacturer wants to make puzzles with at least 1,000 pieces. Among those puzzles she wants the total number of pieces to be as small as possible. She also wants the ratio of the number of columns to the number of rows to be as close as possible to the Golden Ratio. She agrees that minimizing the sum of the percentage deviations from her two targets would be appropriate. Determine the number of rows and columns she should have in her puzzles.
I’d like to find values for \(r\) and \(c\) that minimize \(\frac{|r*c-1000|}{1000} + \frac{\left|c/r - (1+\sqrt{5})/2\right|}{(1+\sqrt{5})/2}\), which is the sum of the percentage deviations from the target of 1000 pieces and the target ratio.
Let’s cycle through several guesses for \(r\) and see which turns out a minimum.
# Define the golden ratio
gr <- (1+sqrt(5))/2
# create a function to minimize
puzzle <- function(r,c){
(abs(r*c - 1000)/1000) + abs(c/r-gr)/gr
}
## Variables to keep track of minimum percentage deviation thus far, and
## the row that goes with it.
PM <- 10
R <- 0
for(r in 20:31){
c <- ceiling(1000/r)
R <- ifelse(puzzle(r,c)<PM, r, R)
PM <- ifelse(puzzle(r,c)< PM, puzzle(r,c), PM)
}
## what is the row that goes with the minimum percentage deviation?
R## [1] 25## and the minimum percentage deviation?
PM## [1] 0.01114562So, a 25X40 jigsaw-puzzle provides exactly 1000 pieces with a 1.6 ratio, a percentage deviation of just over 1%.
A jigsaw puzzle has n edge pieces, where n is an even integer. How many pieces can the puzzle have altogether? For example, if n=14, then the puzzle could have 14, 18, or 20 pieces. (Your answers should be in terms of n.)
When we add the number of columns and rows together we sum each of the four corner pieces twice. So, \(n = 2c+2r-4\). This means that \(\frac{n}{2}+2 = c + r\). From this, we can see that a puzzle could have \(\frac{n}{2}\) columns and 2 rows, \(\frac{n}{2}-1\) columns and 3 rows, etc. until your rows are larger than the number of columns, in which case the puzzle has the same number of pieces.
We’d like a \(k = k(n)\), such that \(\frac{n}{2}-k > 2+k\), which implies that \(\frac{n}{4}-1>k\). Thus, if we let \(k(n) = \lceil \frac{n}{4}-1\rceil\), where \(\lceil x \rceil\) denotes the ceiling of \(x\), we get that there will be \(k(n)+1\) different number of pieces the puzzle could have: \(n, \frac32 n-3, \ldots, (\frac12 n-k(n))(2+k(n))\).
So far in this column, I have written “jigsaw” followed immediately by “puzzle” six times. Half the time I used a hyphen and half the time I didn’t. Provide the reason for hyphen use and non-use.
The title included a hyphen, and then the author alternates the non-use and use of the hyphen. So, my mathematical mind wants to say that the author decided to hyphenate jigsaw puzzle when it was the kth time he used it for odd k, and not hyphenate it for even k.
Instead of jumping to conclusions, I decided to do a little research and found that the three times that it is hyphenated, it is describing something (the first two uses, it describes puzzles, and the last a manufacturer). The times he doesn’t use the hyphen, it is a stand alone noun.
I learned something new. Who would have thought that you could use jigsaw-puzzle as an adjective?