Josephus Nim

Here, we present a variant of Nim with two piles. In the first pile, we have stones with a weight of 1, and in the second pile, we have stones with a weight of -2. Two Players take turns to take stones from one of the piles, and the total weight of stones to be removed should be equal to or less than half of the total weight of the stones in the pile. The player who removed the last stone or stones is the winner of the game. The authors discovered that when (n,m) is a previous player's winning position, 2m+1 is the last remaining number of the Josephus problem, where there are n -numbers, and every second number is to be removed. There are similar relations between the position of which the Grundy number is s and the (n-s)-th removed number

Impartial and Partizan Restricted Chocolate Bar Games

In this paper, we consider impartial and partizan restricted chocolate bar games. In impartial restricted chocolate bar games, players cut a chocolate bar into two pieces along any horizontal or vertical line and eat whichever piece is smaller. If the two pieces are the same size, a player can eat either one. In constrast, partizan restricted chocolate bar games include players designated as Left and Right and chocolate bars with black and white stripes. Left cuts the chocolate bar in two as above and eats the part with fewer black blocks. Similarly, Right cuts the bar and eats the part with fewer white blocks. A player loses when they cannot eat the remaining chocolate bar. We provide formulas that describe the winning positions of the previous player, Right, and Left players. We also present an interesting similarity in the graphs of previous players' winning positions for impartial and partizan chocolate bar games

Games of Nim with Dynamic Restrictions

The authors present formulas for the previous player's winning positions of two variants of restricted Nim. In both of these two games, there is one pile of stones, and in the first variant, we investigate the case that in k-th turn, you can remove f(k) stones at most, where f is a function whose values are natural numbers. In the second variant, there are two kinds of stones. The Type 1 group consists of stones with the weight of one, and the Type 2 group consists of stones with the weight of two. When the total weight of stones is a, you can remove stones whose total weight is equal to or less than half of a

Curious Properties of Iterative Sequences

In this study, several interesting iterative sequences were investigated. First, we define the iterative sequences. We fix function f(n). An iterative sequence starts with a natural number n, and calculates the sequence f(n),f(f(n)), ...f(f(f(f(n)))),... We then search for interesting features in this sequence. We study Kaprekar's routine, the digit factorial process, and the digit power process. The authors presented new variants of Kaprekar's routine.Comment: This is already submitted to a journa

Combination of apodized pupil and phase mask coronagraph for SCExAO at Subaru Telescope

Subaru telescope has been operating a high-contrast imaging instruments called Subaru coronagraphic extreme adaptive optics (SCExAO) which is used for exoplanet research. We are developing phase mask coronagraphs using photonic crystal wave plates inside the SCExAO. An eight-octant phase mask (8OPM) of three-layer achromatic structure has been fabricated as a second generation. It was designed for J and H band to reach 10⁻⁵ contrast, and Ks band to 10⁻⁴. A retardation and a coronagraphic performance of the 8OPM were confirmed almost as designed at 1550nm. An apodised (binary shaped) pupil to be used with the 8OPM was also studied to suppress diffracted light by the secondary shadow and spiders. We confirmed a performance of the combination of the apodizer and the 8OPM at visible wavelengths in a lab. We optimized the apodizer for a pupil of the SCExAO where we obtained a transmission of 50 % and a contrast of 10⁻⁴ the center and 10⁻⁶ at outer region. We manufactured the designed apodizer to be installed in SCExAO for infrared observations

Halftone Wave Front Control: Numerical Simulation and Laboratory Demonstration

High-contrast instruments are required for direct imaging of faint exoplanets around bright host stars. In high-contrast instruments, a wave front control system is needed to generate a dark hole by suppressing residual stellar speckles. However, the achievable contrast is limited by the phase quantization error (i.e., finite phase resolution) of wave front control devices, such as deformable mirrors or spatial light modulators. In this paper, we propose a halftone method for wave front control to improve the contrast using a wave front control device with quantized phase modulation. In a numerical simulation, the contrast was improved from 1.4 x 10(-9) to 3.8 x 10(-10) by halftone wave front control. In addition, we performed a laboratory demonstration in which a spatial light modulator was used for wave front control, and the contrast was improved from 2.2 x 10(-7) to 6.0 x 10(-8) for a phase resolution of 2 pi/256

Synthesis of dihydro-1,4-thiazine from α-keto spiro-thiazolidine

1037-1041The reaction of α-bromocycloalkanone with 2-aminoethanthiol leads to the regioselective formation of spirothiazolidin-2-one with the oxo-group migrating to the original position occupied by the halogen atom. The reaction of 1-thia-4-azaspiro[4.5]alkan-6-one with bromine, iodine, copper (II) salts, acid or base gives dihydro-1,4-thiazine derivatives in moderate yields. Moreover, the treatment of the spiro-thiazolidine derivatives on silica gel under microwave gives the 1,4-thiazine compound