fragmentary ideas ䷿ intellectual what-nots ䷷ and haskell programming ䷴

We don’t usually give much thought to Numeric data types beyond whether we want to work with integers or decimal numbers. And that is a shame! In this post I’ll look at how we can actually do arithmetic operations lazily, and in the process hopefully reveal a bit about haskell’s numeric classes.

We will be using Lennart Augustsson’s numbers library which can be installed from hackage with cabal-install:

$> cabal install numbers

Consider two functions: the Prelude function sum and genericLength from the List library:

 genericLength :: (Num i) => [b] -> i
 sum :: (Num a) => [a] -> a

…two simple functions that have the potential to be very expensive, depending on the length of the list and the values of the elements.

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Note: this is part of a series of posts is related to the “17×17 Challenge” posted by Bill Gasarch. The goal is to color cells of a 17 by 17 grid, using only four colors, such that no rectangle is formed from four cells of the same color.

In my last post, I gave a symmetrical rotation of the known 74-cell single-coloring. I want to use a slight variation of that grid to show why I think treating the grids as symmetrical will help us in solving this problem.

Here is a new spreadsheet, with several panes you can click through on the bottom. Panes ONE through FIVE represent the first few rows of a single-coloring in which we avoid making any real choices, thanks to a few assumptions we make (I’ll come back to that).

Orange represents the row we’re coloring, gray cells are rectangle forming cells (in which marks are not allowed), and blue cells represent what I consider to be the real search-space:

NOTE: We’re simply using another automorphism of the original 74-coloring, i.e. a series of rotations that preserve all the significant attributes of the graph.

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Note: this is part of a series of posts is related to the “17×17 Challenge” posted by Bill Gasarch. The goal is to color cells of a 17 by 17 grid, using only four colors, such that no rectangle is formed from four cells of the same color.

In my last post I noticed that all of the smaller optimal single-colorings I generated showed diagonal symmetry, meaning that row 1 is the same as column 1, row 8 is the same as column 8, etc. It’s my hypothesis that all complete single-colorings are symmetrical in this way.

I decided to play with making the known 74-cell subset symmetrical by applying rotations and came up with this:

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