# Solution of;
# Project Euler Problem 470: Super Ramvok
# https://projecteuler.net/problem=470
# 
# Consider a single game of Ramvok:Let t represent the maximum number of turns 
# the game lasts. If t = 0, then the game ends immediately. Otherwise, on each 
# turn i, the player rolls a die. After rolling, if i < t the player can 
# either stop the game and receive a prize equal to the value of the current 
# roll, or discard the roll and try again next turn. If i = t, then the roll 
# cannot be discarded and the prize must be accepted. Before the game begins, 
# t is chosen by the player, who must then pay an up-front cost ct for some 
# constant c. For c = 0, t can be chosen to be infinite (with an up-front cost 
# of 0). Let R(d, c) be the expected profit (i. e. net gain) that the player 
# receives from a single game of optimally-played Ramvok, given a fair d-sided 
# die and cost constant c. For example, R(4, 0. 2) = 2. 65. Assume that the 
# player has sufficient funds for paying any/all up-front costs. Now consider 
# a game of Super Ramvok:In Super Ramvok, the game of Ramvok is played 
# repeatedly, but with a slight modification. After each game, the die is 
# altered. The alteration process is as follows: The die is rolled once, and 
# if the resulting face has its pips visible, then that face is altered to be 
# blank instead. If the face is already blank, then it is changed back to its 
# original value. After the alteration is made, another game of Ramvok can 
# begin (and during such a game, at each turn, the die is rolled until a face 
# with a value on it appears). The player knows which faces are blank and 
# which are not at all times. The game of Super Ramvok ends once all faces of 
# the die are blank. Let S(d, c) be the expected profit that the player 
# receives from an optimally-played game of Super Ramvok, given a fair d-sided 
# die to start (with all sides visible), and cost constant c. For example, 
# S(6, 1) = 208. 3. Let F(n) = ∑ 4≤d≤n ∑ 0≤c≤n S(d, c). Calculate F(20), 
# rounded to the nearest integer.
# 
# by lcsm29 http://github.com/lcsm29/project-euler
import timed


def dummy(n):
    pass


if __name__ == '__main__':
    n = 1000
    i = 10000
    prob_id = 470
    timed.caller(dummy, n, i, prob_id)

# Solution of;
# Project Euler Problem 470: Super Ramvok
# https://projecteuler.net/problem=470
# 
# Consider a single game of Ramvok:Let t represent the maximum number of turns 
# the game lasts. If t = 0, then the game ends immediately. Otherwise, on each 
# turn i, the player rolls a die. After rolling, if i < t the player can 
# either stop the game and receive a prize equal to the value of the current 
# roll, or discard the roll and try again next turn. If i = t, then the roll 
# cannot be discarded and the prize must be accepted. Before the game begins, 
# t is chosen by the player, who must then pay an up-front cost ct for some 
# constant c. For c = 0, t can be chosen to be infinite (with an up-front cost 
# of 0). Let R(d, c) be the expected profit (i. e. net gain) that the player 
# receives from a single game of optimally-played Ramvok, given a fair d-sided 
# die and cost constant c. For example, R(4, 0. 2) = 2. 65. Assume that the 
# player has sufficient funds for paying any/all up-front costs. Now consider 
# a game of Super Ramvok:In Super Ramvok, the game of Ramvok is played 
# repeatedly, but with a slight modification. After each game, the die is 
# altered. The alteration process is as follows: The die is rolled once, and 
# if the resulting face has its pips visible, then that face is altered to be 
# blank instead. If the face is already blank, then it is changed back to its 
# original value. After the alteration is made, another game of Ramvok can 
# begin (and during such a game, at each turn, the die is rolled until a face 
# with a value on it appears). The player knows which faces are blank and 
# which are not at all times. The game of Super Ramvok ends once all faces of 
# the die are blank. Let S(d, c) be the expected profit that the player 
# receives from an optimally-played game of Super Ramvok, given a fair d-sided 
# die to start (with all sides visible), and cost constant c. For example, 
# S(6, 1) = 208. 3. Let F(n) = ∑ 4≤d≤n ∑ 0≤c≤n S(d, c). Calculate F(20), 
# rounded to the nearest integer.
# 
# by lcsm29 http://github.com/lcsm29/project-euler
import timed


def dummy(n):
    pass


if __name__ == '__main__':
    n = 1000
    i = 10000
    prob_id = 470
    timed.caller(dummy, n, i, prob_id)

# Solution of; # Project Euler Problem 470: Super Ramvok # https://projecteuler.net/problem=470 # # Consider a single game of Ramvok:Let t represent the maximum number of turns # the game lasts. If t = 0, then the game ends immediately. Otherwise, on each # turn i, the player rolls a die. After rolling, if i < t the player can # either stop the game and receive a prize equal to the value of the current # roll, or discard the roll and try again next turn. If i = t, then the roll # cannot be discarded and the prize must be accepted. Before the game begins, # t is chosen by the player, who must then pay an up-front cost ct for some # constant c. For c = 0, t can be chosen to be infinite (with an up-front cost # of 0). Let R(d, c) be the expected profit (i. e. net gain) that the player # receives from a single game of optimally-played Ramvok, given a fair d-sided # die and cost constant c. For example, R(4, 0. 2) = 2. 65. Assume that the # player has sufficient funds for paying any/all up-front costs. Now consider # a game of Super Ramvok:In Super Ramvok, the game of Ramvok is played # repeatedly, but with a slight modification. After each game, the die is # altered. The alteration process is as follows: The die is rolled once, and # if the resulting face has its pips visible, then that face is altered to be # blank instead. If the face is already blank, then it is changed back to its # original value. After the alteration is made, another game of Ramvok can # begin (and during such a game, at each turn, the die is rolled until a face # with a value on it appears). The player knows which faces are blank and # which are not at all times. The game of Super Ramvok ends once all faces of # the die are blank. Let S(d, c) be the expected profit that the player # receives from an optimally-played game of Super Ramvok, given a fair d-sided # die to start (with all sides visible), and cost constant c. For example, # S(6, 1) = 208. 3. Let F(n) = ∑ 4≤d≤n ∑ 0≤c≤n S(d, c). Calculate F(20), # rounded to the nearest integer. # # by lcsm29 http://github.com/lcsm29/project-euler