Largest polyomino with no four cells equally spaced on a straight line
LLargest polyomino with no four cells equallyspaced on a straight line
Jan Kristian Haugland [email protected]
May 5, 2020
Abstract
The maximal number of cells in a polyomino with no four cells equallyspaced on a straight line is determined to be 142. This is based on severalpartial results, each of which can be verified with computer assistance. A polyomino is an object in the plane formed by joining one or more unit squares(called cells ) edge to edge. It can be viewed as a graph with the cells as verticesand an edge joining two vertices if the corresponding cells are adjacent.A polyomino is called admissible if no four cells (i.e., their centres) are equallyspaced on a straight line. A path is a polyomino that either consists of a singlecell, or contains two cells of degree 1 (called endpoints ) while all the remainingcells have degree 2.Suppose a subset of the cells of a polyomino P form a path Q with endpoints A and B . If the graph distance between A and B in P is equal to the graphdiameter d of P , and Q consists of d + 1 cells, then Q is said to be a maximal path in P . The radius of P with respect to Q is then the minimal value of r such that for any cell C in P , there exists a cell D in Q for which the graphdistance between C and D is at most r .A loop is a polyomino for which all the cells have degree 2. The objective of this paper is to outline a verification of the following result:
Theorem 1.
An admissible polyomino may have at most 142 cells.
It is straightforward to generate all admissible paths with a computer pro-gram, and this will serve as the basis of the verification. Suppose, temporarily,that we only considered a special type of paths in which we could traverse the1 a r X i v : . [ m a t h . G M ] M a y ells from one endpoint to the other by always moving either East or North,say, from one cell to the next. Instead of the polyomino itself, we could viewthe path as a sequence on the symbols E, N (East, North). The requirementthat no four cells are equally spaced on a straight line would then be equivalentto requiring that no three consecutive ”blocks” of symbols are permutations ofeach other. Dekking [Dekking, 1979] has shown that such a sequence could beinfinitely long if no four consecutive blocks are permutations of each other, andmentions that the case with three blocks is easily checked to only have finitesolutions. But in our more general case, the paths can have as many as 120cells. The basic idea in verifying Theorem 1 is to go through the admissible paths(or a subset of them, as we shall see later), and for each one, either find thelargest admissible polyominoes that contain it as a maximal path, or find anupper bound for their size.As a polyomino is built one cell at a time from a maximal path, it can beuseful to keep track of the graph diameter of the intermediate polyominoes.Therefore, we start with a result on the existence of loops in large admissiblepolyominoes.
Lemma 1.
An admissible polyomino with at least 67 cells can not contain anyloop of length greater than 4.Verification.
Table 1 shows all admissible loops (up to isometry) of lengthgreater than or equal to 8, together with the maximal number of cells thatmay be added (so that the resulting polyomino remains admissible), and themaximal total number of cells. Each loop is given by a set of directions for mov-ing from one cell to the next around the loop, with E, N, W and S representingEast, North, West and South respectively.Unlike larger loops, a loop of length 4 does not have its graph diameterincreased if a cell is removed. This leads to the following result.
Corollary 1.
We can build any admissible polyomino P with a least 67 cellsby adding one by one cell to a maximal path in P , without altering the graphdiameter at any point. a x . n o . o f M a x . t o t a l L oo p L e n g t h e x t r a ce ll s n o . o f ce ll s EE NN WW SS EE N E NN WW S W SS EE N EE NN W N WW SS W SS EE N E NN W N WW S W SS E S EE N EE NN W N WW S WW SS E S EE N EE NN WW N WW S W SS E S EE N EE NN W NN WW S WW SS E SS EE N EE NN W NN EE N EE NN W NN WW S W S WW N WW S W SS E S E SS W SS E S EE N EE NN W NN EE N EE NN W N WW S WW N W N WW SS W SS E S E SS W SS E S EE N EE NN W NN EE N EE NN W NN WW S W S WW NN WW SS W SS E S E SS W SS E S EE N EE NN W NN EE N EE NN W NN WW SS WW N W N WW SS W SS E S E SS W SS E S EE N EE S E S EE N E NN W NN E N E NN W N WW S WW N W N WW S W SS E SS W S W SS E S EE N EE S E S EE N E NN W N W NN E NN W N WW S WW N W N WW S W SS E S E SS W SS E S EE N EE NN W N W NN E NN W N WW S WW N W N WW S WW SS E S E SS W SS E S EE N EE S E S EE N EE NN WW NN E N E NN W N WW S WW N W N WW S W SS E SS W S W SS E S EE N EE S E S EE N EE NN WW NN E N E NN W NN WW SS WW N W N WW S W SS E SS W S W SS E S EE N EE S E S EE N EE NN WW NN E N E NN W N WW S WW N W N WW S WW SS EE SS W S W SS E S EE N EE S E S EE N EE NN W NN EE N EE NN W N WW S WW N W N WW S WW SS E SS WW S WW SS E S EE N EE S E S EE N EE NN WW NN E N E NN W NN WW SS WW N W N WW S WW SS EE SS W S W SS E S EE N EE S E S EE N EE NN WW NN E N E NN W NN WW SS WW N W N WW S WW SS EE SS W S W SS E SS EE NN EE S E S T a b l e : A d m i ss i b l e l oo p s o f l e n g t h g r e a t e r t h a n o r e q u a l t o8 , a nd t h e m a x i m a l nu m b e r o f e x t r a ce ll s emma 2. If P is an admissible polyomino that does not contain any loop oflength greater than 4, and Q is a maximal path in P , then the radius of P withrespect to Q is at most 5.Verification. There is only one admissible polyomino (up to isometry) that isthe union of three paths of graph diameter 6 that only overlap in one commonendpoint, shown here.However, it contains a loop of length 12, and is not an actual counterexample.If Q is an admissible path, suppose its set of cells is partitioned into one ormore disjoint subsets Q = (cid:91) i =1 ,...,k Q i It seems natural to restrict our attention to the cases in which each Q i is con-nected, although this is not strictly required. Let f ( i ) denote the maximalnumber of cells that can be added to Q by the following iterative steps, assum-ing that a cell can only be added to an admissible polyomino if the resultingpolyomino is also admissible, and if the graph diameter is not altered.Step 1: Add only cells that are adjacent to at least one cell in Q i Step j ∈ { , , ... } : Add only cells that are adjacent to at least onecell that was added in step j − k = 1 and find f (1), the exact number of cells that can be added, butthis can be time consuming. For higher values of k , an upper bound for thenumber of cells that can be added is given by f (1) + f (2) + ... + f ( k )which is often good enough if we are only interested in the global maximum.A combination of the two approaches is also possible: Using upper bounds, wecan determine which paths are good candidates for creating large admissiblepolyominoes, and then we can run a full analysis on those.4 Results
We do not need to go through all paths. For example, it can be verified thatif Q contains 48 cells, and we take the first 16, the middle 16 and the last 16cells as the k = 3 subsets, then we have f (1) ≤ f (2) ≤ f (3) ≤
9. ByLemma 2, this gives us valid upper bounds also for other paths partitioned intosegments of 16 cells in a similar way (i.e., 9 at the ends and 8 everywhere inbetween), and we can cover all diameters from 47 to 90. It has been verifiedthat for any graph diameter less than 106, we have an upper bound for the totalnumber of cells that is less than 142.For each value of the graph diameter from 106 and upwards, the exact max-imal number of cells has been determined.Graph No. of admissible Maximal size of andiameter paths (up to isometry) admissible polyomino119 6 138118 30 138117 55 140116 75 141115 117 142114 144 142113 187 142112 221 141111 266 140110 332 138109 478 136108 679 134107 963 133106 1308 1325ll the admissible polyominoes with 142 cells can be generated by includingexactly one cell of each colour other than black in the figure (provided, of course,that it remains connected), and admissible polyominoes with maximal size forgraph diameters 108 through 112 can be obtained by ”pruning” them.Likewise, the largest admissible polyominoes having the maximal graph di-ameter of 119 can be generated by including exactly one cell of each colour otherthan black in the next figure.
References [Dekking, 1979] Dekking, F. M. (1979). Strongly non-repetitive sequences andprogression-free sets.