Introducing Langtominoes

"Langtominoes" are a physical manifestation of the pairs of numbers as defined in Langford's Problem. [LINK]

The name 'Langtomino' is a mashup of Langford and Pentomino. Term coined by the Author in April, 2025. Maybe should have mashed up with 'polynomino' - yielding Langnomino? They are more like polycubes.... Langcubicals? LoL

Clearly, this page is being developed! 4/2025

A given Langtomino for a given number 'n' has two prongs, separated by n units. Each prong is one unit wide. This is because in Langford's Problem, each number in the arrangement takes up one location, and the pair of n's are separated by n units..

3D view of the 1 LangTomino

Measurements

Here are the dimension for the 3 Langtomino. Could be in inches, half inches, centimeters, meters, miles, etc.

Langtomino dimensions for '3'

This image speaks for itself. View from above, does not show the unit thickness of the pieces.

An arrangement of 8 Langtominoes

Planar Solutions

Here is an early blurb from Knuth, Art of Computer Programming, on the 4 planar solutions for n=8.

This excerpt shows the four planar solutions for 8 pairs

Here are the 4 planar solutions for n=8, as given by Knuth, for your handy text reference: 

1317538642572468

1418634753268257

4275248635713168

5286235743681417

Planarity is not possible unless one utilizes the plane ABOVE and BELOW the arrangement, as in the above diagrams. (I should have a specific page on Planar solutions, but Alas, I don't!)

3D planar solutions?

Let's say you are trying to discover a planar solution, with the pcs flat on the table, but you just couldn't make it work. What if you were allowed to stand a selected Langtomino up (on its two prongs) in the '3D' dimension? That is, Spatially, you'd be using a half plane orthogonal to the solution in progress? That way, you could avoid other langtominoes, while bridging up and over other numbers to connect two vacant positions.

Sorry, no diagram of this action here! Yet.

4D planar solutions!

This is like the 3D solution, in addition, 4D goes down and under to come up in to vacant position.

It seems like 3D and 4D (in addition to 2D) might be able to 'cover' the same number of solutions for Langford's Problem as those where connections cross. I will need to run some experiments to see if this is the case.

Figures

To be clear, Langtominoes would not be flat, paper-thin shapes. They'd be made up of 1" wide 'bars' that are 1" thick. (Or perhaps ½ thick.)

3D-Printed Hexonimoes

I will 3D-Print via ThingVerse or other other site, and then we can have proper images here. I have no timeline on this - hopefully summer 2025 if I can enlist some help.

The Langtomino is a very simple shape. One way to describe one would be as a construction of three rectangular prisms. Another way would be to describe a flat-ish prism with a rectangular bite taken out of it, like this poor rendition:

solid constructive geometry.. a shallow rectangular prism with a chomp taken out of it.

rainbones

Knuth and George Miller made a plastic toy "rainbones" using interlocking arcs, but the pcs have to lie flat on the plane. See link to separate page in references. These were probably molded, rather than printed.

References

A link appears with each reference, to the source.