In: Advanced Math

Alice and Bob are have several piles of chips. On each turn they can either remove 1 or 2 chips from one pile, or split a pile into two nonempty piles. Players take turns and a player that cannot make a move loses. Find the value of the Sprague–Grundy function for positions with one pile made of n chips. (Please do not forget to prove correctness of your asnwer.)

**Nim**
is a combinatorial game, where two players alternately take turns
in taking objects from several heaps. The only rule is that each
player must take at least one object on their turn, but they may
take more than one object in a single turn, as long as they all
come from the same heap.

Nim is the most well-known example of an impartial game, a game where both players have the same moves all the time, and thus the only distinguishing feature is that one player goes first. It is also completely solved, in the sense that the exact strategy has been found for any starting configuration.

Rules

The basic Nim begins
with two players and several *heaps*, each containing
several *objects*. Occasionally, heaps are also called
*piles*, and the objects are called *stones*.

Each player, in turn, must take at least one stone, but they may take more than one stone as long as they all come from the same pile. It's allowed to make a pile empty, effectively removing the pile out of the game. When a player is unable to move, the game ends. Naturally, as long as there is a stone, either player can take that stone, and thus can move. So the ending condition can be rephrased, where the game ends if there is no stone left.

In *normal*
Nim, the loser is the player unable to move. It is called normal
condition by convention from combinatorial game theory, where a
normal game gives the win to the last player making a move. In
*misère* Nim, the player unable to move wins instead; this
is equivalent to the player taking the last stone losing.

Example Game

Consider the following example of the game. There are three piles, initially having 3, 4, 53,4,5 stones respectively. Alice and Bob are playing, with Alice starting.

In normal play, Alice wins, as Alice has taken the last stone and thus leaving Bob with no move. In misère play, Alice would lose instead, but in this case Alice would have taken 2 stones from Pile 3 on move 7, leaving Bob with pile sizes 0, 1, 0 and thus forcing Bob to take the last object.

In the above game, Alice has played a perfect game, never letting Bob to be able to snatch the win. This can be generalized into a general strategy.

Strategy

- f the suggested move makes the pile have 1 stone left, make it have 0 stones instead, or
- If the suggested move makes the pile have 0 stones left, make it have 1 stone instead.

In other words, the correct move is to leave an odd number of piles of size 1. (In normal play, there should be an even number of piles of size 1 instead, to make the nim-sum zero.)

Proof of Winning Strategy

The following proof is for normal Nim's strategy, given by C. Bouton.

theorem:

The moving player wins in normal Nim if and only if the nim-sum of the pile sizes is not zero.

Piles can be fabricated from several different materials.
Compare timber piles to concrete piles, listing the advantages of
each material.

:)
Alice and Bob play the following game: in each round, Alice
first rolls a single standard fair die. Bob then rolls a single
standard fair die. If the difference between Bob’s roll and Alice's
roll is at most one, Bob wins the round. Otherwise, Alice wins the
round.
(a) (5 points) What is the probability that Bob wins a single
round?
(b) (7 points) Alice and Bob play until one of them wins three
rounds. The first player to...

Alice and Bob are playing a game in which each of them has three
strategies, A, B, or C.
If their choices do not match (e.g., if Alice picks B and Bob picks
C), then no money is
exchanged; otherwise Alice pays Bob $6 (if they both choose A), or
$3 (if they both choose
B), or $1 (if they both choose C). Is this a zero-sum game? Find a
mixed-strategy Nash
equilibrium for it. Is this the only equilibrium...

Alice and Bob bet 50 dollars with each other in a game in which
their friend Charlie tosses a two-sided coin 3 times in a remote
location. If Alice correctly predicts the majority face (i.e. the
face which occurred the most often in the three tosses), she gets
to keep Bob's money as well. Charlie calls them and lets them know
that at least 1 heads has occurred.a) Assuming that the coin was
fair, what is the probability that the...

Suppose Alice and Bob have RSA public keys in a file on a
server. They communicate regularly, using authenticated,
confidential message. Eve wants to read the messages but is unable
to crack the RSA private keys of Alice and Bob. However, she is
able to break into the server and alter the file containing Alice’s
and Bob’s public keys.
(1) How should Eve alter the file to so that she can read
confidential messages sent between Alice and Bob, and...

Four witnesses, Alice, Bob, Carol, and Dave, at a trial each
tell the truth with probability 1/3 independent of each other. In
their testimonies, Alice claimed that Bob denied that Carol
declared that Dave lied. What is the conditional probability that
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in c++ please
In this program you are going to have several files to turn in
(NOT JUST ONE!!)
hangman.h – this is your header file – I will
give you a partially complete header file to start with.
hangman.cpp – this is your source file that
contains your main function
functions.cpp – this is your source file that
contains all your other functions
wordBank.txt – this is the
file with words for the game to use. You should put 10...

Suppose that tortilla chips and potato chips are
substitutes goods for each other. Which of the following can cause
the demand for potato chips to fall?a.if potato chips are a normal good, a fall in consumers’
incomesb.if potato chips are a normal good, a rise in consumers’
incomesc.if potato chips are an inferior good, a fall in consumers’
incomesd.none of the abov

we have encountered several firms that have either failed or
came close to collapsing during the Financial Crisis of 2007-2009
(e.g., Bear Stearns, Lehman, AIG, Fannie Mae…).
Select one of these firms and write ：
Describe in some detail the events that led to the failure of
the firm. What went wrong?
Describe the interconnectedness of the firm, and how that may
have played a role in the US government’s response to its pending
failure.

I have this matlab program, and need to turn it into a C++
program. Can anyone help me with this?
% Prompt the user for the values x and y
x = input ('Enter the x coefficient: ');
y = input ('Enter the y coefficient: ');
% Calculate the function f(x,y) based upon
% the signs of x and y.
if x >= 0
if y >= 0
fun = x + y;
else
fun = x + y^2;
end...

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