arXiv subjects

George I. Bell

Publications and source records attributed to George I. Bell.

12 recordsLinked to original sources

arXiv2020

The mathematics of Septoku

Septoku is a Sudoku variant invented by Bruce Oberg, played on a hexagonal grid of 37 cells. We show that up to rotations, reflections, and symbol permutations, there are only six valid Septoku boards. In order to have a unique solution, we show that the minimum number of given values is six. We generalize the puzzle to other board shapes, and devise a puzzle on a star-shaped board with 73 cells with six givens which has a unique solution. We show how this puzzle relates to the unsolved Hadwiger-Nelson problem in combinatorial geometry.

math.CO↗

arXiv2017

Designing peg solitaire puzzles

Peg solitaire is an old puzzle with a 300 year history. We consider two ways a computer can be utilized to find interesting peg solitaire puzzles. It is common for a peg solitaire puzzle to begin from a symmetric board position, we have computed solvable symmetric board positions for four board shapes. A new idea is to search for board positions which have a unique starting jump leading to a solution. We show many challenging puzzles uncovered by this search technique. Clever solvers can take advantage of the uniqueness property to help solve these puzzles.

math.HO↗

arXiv2014

Notes on solving and playing peg solitaire on a computer

We consider the one-person game of peg solitaire played on a computer. Two popular board shapes are the 33-hole cross-shaped board, and the 15-hole triangle board---we use them as examples throughout. The basic game begins from a full board with one peg missing and the goal is to finish at a board position with one peg. First, we discuss ways to solve the basic game on a computer. Then we consider the problem of quickly distinguishing board positions where the goal can still be reached ("winning" board positions) from those where it cannot. This enables a computer to alert the player if a jump under consideration leads to a dead end. On the 15-hole triangle board, it is possible to identify all winning board positions (from any single vacancy start) by storing a key set of 437 board positions. For the "central game" on the 33-hole cross-shaped board, we can identify all winning board positions by storing 839,536 board positions. By viewing a successful game as a traversal of a directed graph of winning board positions, we apply a simple algorithm to count the number of ways to traverse this graph, and calculate that the total number of solutions to the central game is 40,861,647,040,079,968. Our analysis can also determine how quickly we can reach a "dead board position", where a one peg finish is no longer possible.

math.CO↗

arXiv2014

Modulator simulations for coherent electron cooling using a variable density electron beam

Increasing the luminosity of relativistic hadron beams is critical for the advancement of nuclear physics. Coherent electron cooling (CEC) promises to cool such beams significantly faster than alternative methods. We present simulations of 40 GeV/nucleon Au+79 ions through the first (modulator) section of a coherent electron cooler. In the modulator, the electron beam copropagates with the ion beam, which perturbs the electron beam density and velocity via anisotropic Debye shielding. In contrast to previous simulations, where the electron density was constant in time and space, here the electron beam has a finite transverse extent, and undergoes focusing by quadrupoles as it passes through the modulator. The peak density in the modulator increases by a factor of 3, as specified by the beam Twiss parameters. The inherently 3D particle and field dynamics is modeled with the parallel VSim framework using a $δ$f PIC algorithm. Physical parameters are taken from the CEC proof-of-principle experiment under development at Brookhaven National Lab.

physics.acc-ph↗

arXiv2012

Fringe field simulations of a non-scaling FFAG accelerator

Fixed-field Alternating Gradient (FFAG) accelerators offer the potential of high-quality, moderate energy ion beams at low cost. Modeling of these structures is challenging with conventional beam tracking codes because of the large radial excursions of the beam and the significance of fringe field effects. Numerous tune resonances are crossed during the acceleration, which would lead to beam instability and loss in a storage ring. In a non-scaling FFAG, the hope is that these resonances can be crossed sufficiently rapidly to prevent beam loss. Simulations are required to see if this is indeed the case. Here we simulate a non-scaling FFAG which accelerates protons from 31 to 250 MeV. We assume only that the bending magnets have mid-plane symmetry, with specified vertical bending field in the mid-plane (y=0). The magnetic field can be obtained everywhere using a power series expansion, and we develop mathematical tools for calculating this expansion to arbitrary order when the longitudinal field profile is given by an Enge function. We compare the use of a conventional hard-edge fringe with a more accurate, soft-edge fringe field model. The tune 1/3 resonance is the strongest, and crossing it in the hard-edge fringe model results in a 21% loss of the beam. Using the soft-edge fringe model the beam loss is less than 6%.

physics.acc-ph↗

arXiv2009

Solving Triangular Peg Solitaire

We consider the one-person game of peg solitaire on a triangular board of arbitrary size. The basic game begins from a full board with one peg missing and finishes with one peg at a specified board location. We develop necessary and sufficient conditions for this game to be solvable. For all solvable problems, we give an explicit solution algorithm. On the 15-hole board, we compare three simple solution strategies. We then consider the problem of finding solutions that minimize the number of moves (where a move is one or more consecutive jumps by the same peg), and find the shortest solution to the basic game on all triangular boards with up to 55 holes (10 holes on a side).

math.CO↗

arXiv2009

The shortest game of Chinese Checkers and related problems

In 1979, David Fabian found a complete game of two-person Chinese Checkers in 30 moves (15 by each player) [Martin Gardner, Penrose Tiles to Trapdoor Ciphers, MAA, 1997]. This solution requires that the two players cooperate to generate a win as quickly as possible for one of them. We show, using computational search techniques, that no shorter game is possible. We also consider a solitaire version of Chinese Checkers where one player attempts to move her pieces across the board in as few moves as possible. In 1971, Octave Levenspiel found a solution in 27 moves [Ibid.]; we demonstrate that no shorter solution exists. To show optimality, we employ a variant of A* search, as well as bidirectional search.

math.CO↗

arXiv2008

Triangular Peg Solitaire Unlimited

Triangular peg solitaire is a well-known one-person game or puzzle. When one peg captures many pegs consecutively, this is called a sweep. We investigate whether the game can end in a dramatic fashion, with one peg sweeping all remaining pegs off the board. For triangular boards of side 6 and 8 (with 21 and 36 holes, respectively) the geometrically longest sweep can occur as the final move in a game. On larger triangular boards, we demonstrate how to construct solutions that finish with arbitrarily long sweeps. We also consider the problem of finding solutions that minimize the total number of moves (where a move is one or more consecutive jumps by the same peg).

math.CO↗

arXiv2007

Diamond Solitaire

We investigate the game of peg solitaire on different board shapes, and find those of diamond or rhombus shape have interesting properties. When one peg captures many pegs consecutively, this is called a sweep. Rhombus boards of side 6 have the property that no matter which peg is missing at the start, the game can be solved to one peg using a maximal sweep of length 16. We show how to construct a solution on a rhombus board of side 6i, where the final move is a maximal sweep of length r, where r=(9i-1)(3i-1) is a "rhombic matchstick number".

math.CO↗

arXiv2007

The minimum size required of a solitaire army

The solitaire army is a one-person peg jumping game where a player attempts to advance an "army" of pegs as far as possible into empty territory. The game was introduced by John Conway and is also known as "Conway's Soldiers". We consider various generalizations of this game in different 2D geometries, unify them under a common mathematical framework, and find the minimum size army capable of advancing a given number of steps.

math.CO↗

arXiv2007

Diagonal Peg Solitaire

We study the classical game of peg solitaire when diagonal jumps are allowed. We prove that on many boards, one can begin from a full board with one peg missing, and finish with one peg anywhere on the board. We then consider the problem of finding solutions that minimize the number of moves (where a move is one or more jumps by the same peg), and find the shortest solution to the "central game", which begins and ends at the center. In some cases we can prove analytically that our solutions are the shortest possible, in other cases we apply A* or bidirectional search heuristics.

math.CO↗

arXiv2006

New problems on old solitaire boards

Some old peg solitaire boards are brought down from the literature, dusted off, and re-examined, and some remarkable problems are displayed on them.

math.CO↗