Ted Harding posed an interesting puzzle challenge on the r-help mailing list recently. Here's the puzzle:
Take the numbers 1, 2, 3, etc. up to 17.
Can you write out all seventeen numbers in a line so that every pair of numbers that are next to each other, adds up to give a square number?
You can figure out the solution from first principles fairly easily (hint: what neighbours must 17 have?), but Ted's challenge was to write a “neat” R function to solve this puzzle programmatically. The r-help thread generated several solutions (including an elegant one based on recursion, and a generalization to larger sets of numbers), but I wanted to highlight this solution from Vincent Zoonekynd on StackOverflow:
# Allowable pairs form a graph p <- outer( 1:17, 1:17, function(u,v) round(sqrt(u + v),6) == floor(sqrt(u+v)) ) ) rownames(p) <- colnames(p) <- 1:17 image(p, col=c(0,1)) # Read the solution on the plot library(igraph) g <- graph.adjacency(p, "undirected") V(g)$label <- V(g)$name plot(g, layout=layout.fruchterman.reingold)
It's a clever use of the graph theory. Consider the numbers 1-17 as vertices in a graph, with connections between pairs defined by whether they sum to a square. The call to outer defines the T/F adjacency matrix (which the code displays as an image), and then the plot.igraph function displays the graph as an R chart:
StackOverflow: Ordering 1:17 by perfect square pairs