A game tree is a rooted tree whose nodes are possible histories of play and whose edges are legal next actions. Internal nodes specify the acting player or a chance move; terminal nodes carry outcomes or payoffs. In an extensive-form game, information sets additionally say which decision histories the acting player can distinguish.

Distinct histories may reach the same board position, so a game tree can contain repeated states that a state graph would merge. For finite two-player zero-sum games with perfect information, minimax search propagates terminal values upward. Chance nodes require expectations instead of maximizing or minimizing their outcomes.

A stochastic game combines a state space with strategic choices and probabilistic state transitions. At each stage, the current state and players' actions determine immediate payoffs and a probability distribution for the next state. A discounted, average-reward or finite-horizon rule specifies how stage rewards are aggregated.

With a single decision maker this reduces to a Markov decision process. With several players, a policy that is best against one fixed opponent may cease to be best when that opponent changes policy. A two-player zero-sum game discounted model leads to a statewise minimax Bellman equation, connecting strategic reasoning with dynamic programming.

The Tower of Hanoi has three pegs and $n$ differently sized discs. Start with all discs stacked largest below smallest on one peg; move one exposed disc at a time, never placing a larger disc on a smaller one, and transfer the stack to another peg.

To move the largest disc, the upper $n-1$ discs must first be collected on the third peg. After its move they must be transferred again. Hence the minimum satisfies $T(n)=2T(n-1)+1$, $T(0)=0$, giving $T(n)=2^n-1$. The recursive construction attains this lower bound. The problem shows how a short recursive procedure can require exponentially many executed moves.

A mathematical game specifies participants, available choices, information and consequences of combined choices. A normal-form game records complete strategies and payoffs; an extensive-form game records the sequence of decisions and what each player knows at them.

For example, matching pennies gives each player two actions and opposing payoffs depending on whether their choices agree. Its equilibrium requires mixed strategies. A puzzle with one decision maker is often called a game informally, but strategic interaction and game theory enter when outcomes depend on several decision makers' choices. The rules determine the mathematical object; a physical board alone does not.

A pagoda function assigns a weight to each location in a move puzzle so that the total weight of occupied locations never increases under a legal move. Comparing the initial and desired totals can then prove that the desired position is unreachable.

In peg solitaire, a jump removes pegs at $a,b$ and adds one at $c$. Weights satisfying $w(c)\le w(a)+w(b)$ for every allowed jump make $P={\sum }_{\text{occupied }x}w(x)$ nonincreasing. If a target has larger $P$ than the initial board, no move sequence can reach it. Finding suitable weights is the creative step; failure of one chosen weight function does not prove reachability.

A partizan game gives the two players, traditionally Left and Right, potentially different legal moves from the same position. Its recursive notation is $G=\{G^L\mid G^R\}$, listing Left's and Right's options. An impartial game is the special case where those option sets agree at every reachable position.

For example $1=\{0\mid \}$ lets Left move to zero while Right has no move. The game $-1=\{\mid 0\}$ reverses the roles, and their disjunctive sum is zero in game value: responding in the other component leaves the previous player winning. Such comparisons concern behavior in sums, rather than simply counting moves available to one player.

Combinatorial game theory studies positions in alternating-move games with fully specified legal moves, usually perfect information and no chance. A position can be defined recursively by the positions available to each player. Under the normal-play convention, a player with no legal move loses.

An impartial game gives both players the same moves; Nim is the standard example. A partizan game allows different moves for Left and Right. The disjunctive sum lets a player move in exactly one component on each turn, making it possible to study how game positions combine algebraically. Infinite play or misère play requires additional conventions; normal-play results do not automatically transfer.

A multi-type Galton-Watson process assigns each individual one of finitely many types. A type-$i$ individual independently produces a random vector of offspring counts, with a distribution depending only on its type. If $Z_n$ is a row vector of generation counts and $M_{ij}$ is the expected number of type-$j$ children of a type-$i$ parent, then $\mathbb{E}[Z_{n+1}\mid Z_n]=Z_{n}M$. Vector probability generating functions iterate to give generation laws and extinction probabilities. The spectral radius of $M$ replaces the scalar offspring mean in irreducible, nonsingular finite-type extinction criteria; reducible systems need componentwise analysis. This models populations with distinct inherited or changing categories.

A zone-centre effective-mass description retaining the s-like conduction band, heavy-hole and light-hole branches of the p-like $j=3/2$ valence manifold, and the spin-orbit-split $j=1/2$ valence branch. Each named branch includes spin multiplicity; four bands does not mean four individual quantum states.

The circumgalactic medium is gas surrounding a galaxy's main stellar body and interstellar gas, typically associated with its halo. accretion, outflow and recycling exchange matter through it. Its observational boundary and inferred mass depend on the adopted halo convention and the gas phases sampled by absorption or emission.

NGC 4258 hosts a resolved nuclear water-maser disk used as an independent nearby-galaxy distance anchor through orbital dynamics.^[ref]

The Large Magellanic Cloud is a nearby galaxy interacting with the Milky Way, containing stars whose independently measurable properties help calibrate astronomical distances. Its proximity makes it possible to combine stellar observations that are much harder to separate in more distant systems.

In a late-type eclipsing binary, orbital velocities and eclipse geometry constrain linear stellar sizes; a surface-brightness calibration supplies corresponding angular sizes. Comparing the two gives a distance anchor rather than assuming one fixed luminosity for every star.^[ref] Such calibration still has systematic uncertainty and samples a galaxy with finite depth.

A diacylglycerol with one galactose headgroup, often abbreviated MGDG.

A diacylglycerol with a digalactosyl headgroup, often abbreviated DGDG. Phosphate deprivation can increase its abundance in extraplastidial membranes.

A mutation giving a qualitatively altered activity, such as a new substrate reaction. Mutant IDH1 production of 2-hydroxyglutarate provides an example.

A gadolinium isotope with a very large thermal-neutron absorption cross section. Its isotope-specific cross section is not the same as the isotope-averaged value for natural gadolinium.