Hardness of Mastermind

Mastermind is a popular board game released in 1971, where a codemaker chooses a secret pattern of colored pegs, and a codebreaker has to guess it in several trials. After each attempt, the codebreaker gets a response from the codemaker containing some information on the number of correctly guessed pegs. The search space is thus reduced at each turn, and the game continues until the codebreaker is able to find the correct code, or runs out of trials. In this paper we study several variations of #MSP, the problem of computing the size of the search space resulting from a given (possibly fictitious) sequence of guesses and responses. Our main contribution is a proof of the #P-completeness of #MSP under parsimonious reductions, which settles an open problem posed by Stuckman and Zhang in 2005, concerning the complexity of deciding if the secret code is uniquely determined by the previous guesses and responses. Similarly, #MSP stays #P-complete under Turing reductions even with the promise that the search space has at least k elements, for any constant k. (In a regular game of Mastermind, k=1.) All our hardness results hold even in the most restrictive setting, in which there are only two available peg colors, and also if the codemaker's responses contain less information, for instance like in the so-called single-count (black peg) Mastermind variation.Comment: 12 pages; Sixth International Conference on FUN WITH ALGORITHMS, 201

Effect of the over-ageing treatment on the mechanical properties of AA2024 aluminum alloy.

The evolution of the hardness of the over-ageing AA2024 alloy scale was followed by measurements of Vickers hardness. The nanoindentation is adapted to the determination of elastoplastic properties (hardness and Young’s modulus) of the matrix and also of coarse intermetallic precipitates. Influence of the artificial over-ageing time to hardness and to mechanical properties as the local scale was investigated

A new approach to local hardness

The applicability of the local hardness as defined by the derivative of the chemical potential with respect to the electron density is undermined by an essential ambiguity arising from this definition. Further, the local quantity defined in this way does not integrate to the (global) hardness - in contrast with the local softness, which integrates to the softness. It has also been shown recently that with the conventional formulae, the largest values of local hardness do not necessarily correspond to the hardest regions of a molecule. Here, in an attempt to fix these drawbacks, we propose a new approach to define and evaluate the local hardness. We define a local chemical potential, utilizing the fact that the chemical potential emerges as the additive constant term in the number-conserving functional derivative of the energy density functional. Then, differentiation of this local chemical potential with respect to the number of electrons leads to a local hardness that integrates to the hardness, and possesses a favourable property; namely, within any given electron system, it is in a local inverse relation with the Fukui function, which is known to be a proper indicator of local softness in the case of soft systems. Numerical tests for a few selected molecules and a detailed analysis, comparing the new definition of local hardness with the previous ones, show promising results.Comment: 30 pages (including 6 figures, 1 table

Short-term hot hardness characteristics of rolling-element steels

Short-term hot hardness studies were performed with five vacuum-melted steels at temperatures from 294 to 887 K (70 to 1140 F). Based upon a minimum Rockwell C hardness of 58, the temperature limitation on all materials studied was dependent on the initial room temperature hardness and the tempering temperature of each material. For the same room temperature hardness, the short-term hot hardness characteristics were identical and independent of material composition. An equation was developed to predict the short-term hardness at temperature as a function of initial room temperature hardness for AISI 52100, as well as the high-speed tool steels

The Effect of Heat Treatment on Mechanical Properties of Cu-0.93 mass％Ni-0.24 mass％P Alloy

Cu-Ni-P alloys are the typical precipitation-hardening material. Thus, Cu-Ni-P alloys are expected to apply alternative materials for heat-exchanger tubes. Therefore, it was investigated that the effect of isothermal artificial aging and the heat-treatment simulated brazing on mechanical properties of Cu-Ni-P alloy in this study. From Vickers hardness test for each artificial aging time, the peak hardness after aging for 10.8 ks achieved at approximately 130 HV. Furthermore, tensile strength of the peak hardness specimen was shown approximately 250 MPa, which was approximately 60 MPa higher than the specimen as solution heat treated. The nominal strain till fracture with isothermal artificial aging was almost equal to the specimen as solution heat treated. These results suggest saving energy in production process for heat exchangers. Spherical or circular precipitates with 5-10 nm diameters were observed in a material which exhibited the peak hardness by transmission electron microscopy. Furthermore, the hardness of specimen as heat-treatment simulated brazing after solution heat treated was shown approximately 105 HV. The peak hardness and tensile strength of the specimen as performed aged at 498 K for 43.2 ks following with the heat-treatment simulated brazing achieved at approximately 140 HV and 260 MPa, respectively. From these results, it is expected the application to higher-strength tube for heat-exchanger after brazing in a furnace