George Flanagin

Software & other things

Tiling and Quilt patterns based on Knuth's Leaper

To get a feel for it, set "Segment 1" to 1, "Segment 2" to 8 and draw your tiling on a 41 x 51 floor. Press the (Draw it!) button. Now increase "Segment 1" to 3, 5, 7, 9, ... up to 25 and see how the pattern changes.

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Note: The max size is 100 x 100. It is a limitation of my eyesight rather than the software.

Ideas & Suggestions: If you are unfamiliar with the words 'relatively prime', segment lengths that differ by 1 are always relatively prime, and you cannot have segments that are both even or both odd and cover all the squares. Other restrictions apply.

Donald Knuth suggests that "Fibonnaci" segments like (5,8) and (8,13) make for interesting patterns, and he suggests them as topics for research. His suggestion is a good starting point. Small segments such as (3,4) or (5,6) on large surfaces make for a familiar wave-like appearance. Also of interest are widely separated segments such as (2,11) and (3,14) on surfaces that are just big enough. Segments that differ by the minimum amounts such as (11,12) and (11,14) also make for some very jumbled patterns on surfaces that are a tight fit.

Pattern Parameters:


Segment 1:

Segment 2:

Tiles in the X dim:

Tiles in the Y dim:

Note: The segment lengths must be chosen so that their sum and difference have no common factor, and the surface must be at least as wide as the sum of the segments, and at least as long as twice the longer segment. But don't worry, the program will help you if you make a mistake.

Types of different tiles needed:


This is a graphic solution to Knuth's problem stated in the addendum to "Leaper Graphs" in The Mathematical Gazette Issue 78, 1994, pp 274-297.


There is a long history in math of research into tiling patterns (tessellations). For many quilts, floors and patios, all we really need is an interesting pattern that does not repeat over some roughly rectangular area.

Full history:

If you need or want a little more background, here is my page on the subject.