{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# The Devil and the Coin Flip Game\n",
"\n",
"If the Devil ever challenges me to a [fiddle contest](https://en.wikipedia.org/wiki/The_Devil_Went_Down_to_Georgia), I'm going down. But here is a contest where I'd have a better chance:\n",
"\n",
"> *You're playing a game with the Devil, with your soul at stake. You're sitting at a circular table which has 4 coins, arranged in a diamond, at the 12, 3, 6, and 9 o'clock positions. You are blindfolded, and can never see the coins or the table.*\n",
"\n",
"> *Your goal is to get all 4 coins showing heads, by telling the devil the position(s) of some coins to flip. We call this a \"move\" on your part. The Devil must faithfully perform the requested flips, but may first sneakily rotate the table any number of quarter-turns, so that the coins are in different positions. You keep making moves, and the Devil keeps rotating and flipping, until all 4 coins show heads.*\n",
"\n",
"> *Example: You tell the Devil to flip the 12 o'clock and 6 o'clock positions. The devil might rotate the table a quarter turn clockwiae, and then flip the coins that have moved into the 12 o'clock and 6 o'clock positions (which were formerly at 3 o'clock and 9 o'clock). Or the Devil could have made any other rotation before flipping.*\n",
"\n",
"> *What is a shortest sequence of moves that is **guaranteed** to win, no matter what the initial state of the coins, and no matter what rotations the Devil applies?*\n",
"\n",
"# Analysis\n",
"\n",
"- We're looking for a \"shortest sequence of moves\" that reaches a goal. That's a [shortest path search problem](https://en.wikipedia.org/wiki/Shortest_path_problem). I've done that before.\n",
"- Since the Devil gets to make moves too, you might think that this is a [minimax](https://en.wikipedia.org/wiki/Minimax) problem: that we should choose the move that leads to the shortest path, given that the Devil has the option of making moves that lead to the longest path.\n",
"- But minimax only works when you know what moves the opponent is making: he did *that*, so I'll do *this*. In this problem the player is blinfolded; that makes it a [partially observable problem](https://en.wikipedia.org/wiki/Partially_observable_system) (in this case, not observable at all, but we still say \"partially\").\n",
"- In such problems, we don't know for sure the true state of the world before or after any move. So we should represent what *is* known: *the set of states that we believe to be possible*. We call this a *belief state*. At the start of the game, each of the four coins could be either heads or tails, so that's 24 = 16 possibilities in the initial belief state:\n",
" {HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, \n",
" THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT}\n",
"- So we have a single-agent shortest path search in the space of belief states (not the space of physical states of the coins). We search for a path from the inital belief state to the goal belief state, which is `{HHHH}`.\n",
"- A move updates the belief state as follows: for every four-coin sequence in the current belief state, rotate it in every possible way, and then flip the coins specified by the position(s) in the move. Collect all these results together to form the new belief state. The search space is small (just 216 possible belief states), so run time will be fast. \n",
"- I'll Keep It Simple, and not worry about rotational symmetry (although we'll come back to that later).\n",
"\n",
"\n",
"# Basic Data Structures and Functions\n",
"\n",
"What data structures will I be dealing with?\n",
"\n",
"- `Coins`: a *coin sequence* (four coins, in order, on the table) is represented as a `str` of four characters, such as `'HTTT'`. \n",
"- `Belief`: a *belief state* is a `frozenset` of `Coins` (frozen so it can be hashed), like `{'HHHT', 'TTTH'}`.\n",
"- `Position`: an integer index into the coin sequence; position `0` selects the `H` in `'HTTT'`\n",
"and corresponds to the 12 o'clock position; position 1 corresponds to 3 o'clock, and so on.\n",
"- `Move`: a set of positions to flip, such as `{0, 2}`. \n",
"- `Strategy`: an ordered list of moves. A\n",
"blindfolded player has no feedback, there thus no decision points in the strategy. "
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from collections import deque\n",
"from itertools import product, combinations, chain\n",
"import random\n",
"\n",
"Coins = ''.join # A coin sequence; a str: 'HHHT'.\n",
"Belief = frozenset # A set of possible coin sequences: {'HHHT', 'TTTH'}.\n",
"Position = int # An index into a coin sequence.\n",
"Move = set # A set of positions to flip: {0, 2}\n",
"Strategy = list # A list of Moves: [{0, 1, 2, 3}, {0, 2}, ...]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What basic functions do I need to manipulate these data structures?\n",
"\n",
"- `all_moves()`: returns a list of every possible move a player can make.\n",
"- `all_coins()`: returns a belief state consisting of the set of all 16 possible coin sequences: `{'HHHH', 'HHHT', ...}`.\n",
"- `rotations(coins)`: returns a belief set of all 4 rotations of the coin sequence.\n",
"- `flip(coins, move)`: flips the specified positions within the coin sequence.\n",
" (But leave `'HHHH'` alone, because it ends the game.)\n",
"- `update(belief, move)`: returns an updated belief state: all the coin sequences that could result from any rotation followed by the specified flips.\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def all_moves() -> [Move]: \n",
" \"List of all possible moves.\"\n",
" return [set(m) for m in powerset(range(4))]\n",
"\n",
"def all_coins() -> Belief:\n",
" \"The belief set consisting of all possible coin sequences.\"\n",
" return Belief(map(Coins, product('HT', repeat=4)))\n",
"\n",
"def rotations(coins) -> {Coins}: \n",
" \"A set of all possible rotations of a coin sequence.\"\n",
" return {coins[r:] + coins[:r] for r in range(4)}\n",
"\n",
"def flip(coins, move) -> Coins:\n",
" \"Flip the coins in the positions specified by the move (but leave 'HHHH' alone).\"\n",
" if 'T' not in coins: return coins # Don't flip 'HHHH'\n",
" coins = list(coins) # Need a mutable sequence\n",
" for i in move:\n",
" coins[i] = ('H' if coins[i] == 'T' else 'T')\n",
" return Coins(coins)\n",
"\n",
"def update(belief, move) -> Belief:\n",
" \"Update belief: consider all possible rotations, then flip.\"\n",
" return Belief(flip(c, move)\n",
" for coins in belief\n",
" for c in rotations(coins))\n",
"\n",
"flatten = chain.from_iterable\n",
"\n",
"def powerset(iterable): \n",
" \"powerset([1,2,3]) --> () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)\"\n",
" # https://docs.python.org/3/library/itertools.html#itertools-recipes\n",
" s = list(iterable)\n",
" return flatten(combinations(s, r) for r in range(len(s) + 1))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's try out these functions:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[set(),\n",
" {0},\n",
" {1},\n",
" {2},\n",
" {3},\n",
" {0, 1},\n",
" {0, 2},\n",
" {0, 3},\n",
" {1, 2},\n",
" {1, 3},\n",
" {2, 3},\n",
" {0, 1, 2},\n",
" {0, 1, 3},\n",
" {0, 2, 3},\n",
" {1, 2, 3},\n",
" {0, 1, 2, 3}]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"all_moves()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"frozenset({'HHHH',\n",
" 'HHHT',\n",
" 'HHTH',\n",
" 'HHTT',\n",
" 'HTHH',\n",
" 'HTHT',\n",
" 'HTTH',\n",
" 'HTTT',\n",
" 'THHH',\n",
" 'THHT',\n",
" 'THTH',\n",
" 'THTT',\n",
" 'TTHH',\n",
" 'TTHT',\n",
" 'TTTH',\n",
" 'TTTT'})"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"all_coins()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'HHHT', 'HHTH', 'HTHH', 'THHH'}"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rotations('HHHT')"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'THTT'"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"flip('HHHT', {0, 2})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are 16 coin sequences in the `all_coins` belief state. If we update this belief state by flipping all 4 positions, we should get a new belief state where we have eliminated the possibility of 4 tails, leaving 15 possible coin sequences:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"frozenset({'HHHH',\n",
" 'HHHT',\n",
" 'HHTH',\n",
" 'HHTT',\n",
" 'HTHH',\n",
" 'HTHT',\n",
" 'HTTH',\n",
" 'HTTT',\n",
" 'THHH',\n",
" 'THHT',\n",
" 'THTH',\n",
" 'THTT',\n",
" 'TTHH',\n",
" 'TTHT',\n",
" 'TTTH'})"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"update(all_coins(), {0, 1, 2, 3})"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(), (1,), (2,), (3,), (1, 2), (1, 3), (2, 3), (1, 2, 3)]"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"list(powerset([1,2,3]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Everything looks good so far. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Search for a Winning Strategy\n",
"\n",
"The generic function `search` does a breadth-first search starting\n",
"from a `start` state, looking for a `goal` state, considering possible `actions` at each turn,\n",
"and computing the `result` of each action (`result` is a function such that `result(state, action)` returns the new state that results from executing the action in the current state). `search` works by keeping a `queue` of unexplored possibilities, where each entry in the queue is a pair consisting of a *strategy* (sequence of moves) and a *state* that that strategy leads to. We also keep track of a set of `explored` states, so that we don't repeat ourselves. I've defined this function (or one just like it) multiple times before, for use in different search problems."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def search(start, goal, actions, result) -> Strategy:\n",
" \"Breadth-first search from start state to goal; return strategy to get to goal.\"\n",
" explored = set()\n",
" queue = deque([(Strategy(), start)])\n",
" while queue:\n",
" (strategy, state) = queue.popleft()\n",
" if state == goal:\n",
" return strategy\n",
" for action in actions:\n",
" state2 = result(state, action)\n",
" if state2 not in explored:\n",
" queue.append((strategy + [action], state2))\n",
" explored.add(state2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that `search` doesn't know anything about belief states—it is designed to work on plain-old physical states of the world. But amazingly, we can still use it to search over belief states: it just works, as long as we properly specify the start state, the goal state, and the means of moving between states.\n",
"\n",
"The `coin_search` function calls `search` to solve our specific problem:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[{0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3}]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def coin_search() -> Strategy: \n",
" \"Use `search` to solve the Coin Flip problem.\"\n",
" return search(start=all_coins(), goal={'HHHH'}, actions=all_moves(), result=update)\n",
"\n",
"coin_search()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"That's a 15-move strategy that is guaranteed to lead to a win. **Stop here** if all you want is the answer to the puzzle. \n",
"\n",
"----\n",
"\n",
"Or you can continue on ...\n",
"\n",
"# Verifying the Winning Strategy\n",
"\n",
"I don't have a proof, but I have some evidence that this strategy works:\n",
"- Exploring with paper and pencil, it looks good. \n",
"- A colleague did the puzzle and got the same answer. \n",
"- It passes the `winning` test below.\n",
"\n",
"The call `winning(strategy, k)` plays the strategy *k* times against a Devil that chooses starting positions and rotations at random. Note this is dealing with concrete, individual states of the world, like `HTHH`, not belief states. If `winning` returns `False`, then the strategy is *definitely* flawed. If it returns `True`, then the strategy is *probably* good—it won *k* times in a row—but that does not prove it will win every time (and either way `winning` makes no claims about being a *shortest* strategy)."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"True"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def winning(strategy, k=100000) -> bool:\n",
" \"Is this a winning strategy? A probabilistic algorithm.\"\n",
" return all(play(strategy) == 'HHHH'\n",
" for _ in range(k))\n",
"\n",
"def play(strategy, starting_coins=list(all_coins())) -> Coins:\n",
" \"Play strategy for one game against a random Devil; return final state of coins.\"\n",
" coins = random.choice(starting_coins)\n",
" for move in strategy:\n",
" if 'T' not in coins: return coins\n",
" coins = random.choice(list(rotations(coins)))\n",
" coins = flip(coins, move)\n",
" return coins\n",
"\n",
"winning(strategy=coin_search())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Canonical Coin Sequences\n",
"\n",
"Consider these coin sequences: `{'HHHT', 'HHTH', 'HTHH', 'THHH'}`. In a sense, these are all the same: they all denote the same sequence of coins with the table rotated to different degrees. Since the devil is free to rotate the table any amount at any time, we could be justified in treating all four of these as equivalent, and collapsing them into one representative member. I will redefine `Coins` so that is stil takes an iterable of `'H'` or `'T'` characters and joins them into a `str`, but I will make it consider all possible rotations of the string and (arbitraily) choose the one that comes first in alphabetical order (which would be `'HHHT'` for the four coin sequences mentioned here)."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"def Coins(coins) -> str: \n",
" \"The canonical representation (after rotation) of the 'H'/'T' sequence.\"\n",
" return min(rotations(''.join(coins)))"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"assert Coins('HHHT') == Coins('HHTH') == Coins('HTHH') == Coins('THHH') == 'HHHT'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With `Coins` redefined, the result of `all_coins()` is different:"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"frozenset({'HHHH', 'HHHT', 'HHTT', 'HTHT', 'HTTT', 'TTTT'})"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"all_coins()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The starting belief set is down from 16 to 6, namely 4 heads, 3 heads, 2 adjacent heads, 2 opposite heads, 1 head, and no heads, respectively. \n",
"\n",
"Let's make sure we didn't break anything:"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[{0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3}]"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"coin_search()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Winning Strategies for *N* Coins\n",
"\n",
"What if there are 3 coins on the table arranged in a triangle? Or 6 coins in a hexagon? To answer that, I'll generalize all the functions that have a \"4\" in them: `all_moves, all_coins`, `rotations` and `coin_search`.\n",
"\n",
"Computing `all_coins(N)` is easy; just take the product, and canonicalize each one with the new definition of `Coins`. For `all_moves(N)` I want canonicalized moves: the moves `{0}` and `{1}` should be considered the same, since they both say \"flip one coin.\" To get that, look at the canonicalized set of `all_coins(N)`, and for each one pull out the set of positions that have an `H` in them and flip those positions. (The positions with a `T` should be symmetric, so we don't need them as well.)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def all_moves(N=4) -> [Move]:\n",
" \"All canonical moves for a sequence of N coins.\"\n",
" return [set(i for i in range(N) if coins[i] == 'H')\n",
" for coins in sorted(all_coins(N))]\n",
"\n",
"def all_coins(N=4) -> Belief:\n",
" \"Return the belief set consisting of all possible coin sequences.\"\n",
" return Belief(map(Coins, product('HT', repeat=N)))\n",
"\n",
"def rotations(coins) -> {Coins}: \n",
" \"A list of all possible rotations of a coin sequence.\"\n",
" return {coins[r:] + coins[:r] for r in range(len(coins))}\n",
"\n",
"def coin_search(N=4) -> Strategy: \n",
" \"Use the generic `search` function to solve the Coin Flip problem.\"\n",
" return search(start=all_coins(N), goal={'H' * N}, actions=all_moves(N), result=update)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's test the new definitions and make sure we haven't broken `update` and `coin_search`:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"assert all_moves(3) == [{0, 1, 2}, {0, 1}, {0}, set()]\n",
"assert all_moves(4) == [{0, 1, 2, 3}, {0, 1, 2}, {0, 1}, {0, 2}, {0}, set()]\n",
"\n",
"assert all_coins(4) == {'HHHH', 'HHHT', 'HHTT', 'HTHT', 'HTTT', 'TTTT'}\n",
"assert all_coins(5) == {'HHHHH','HHHHT', 'HHHTT','HHTHT','HHTTT', 'HTHTT', 'HTTTT', 'TTTTT'}\n",
"\n",
"assert rotations('HHHHHT') == {'HHHHHT', 'HHHHTH', 'HHHTHH', 'HHTHHH', 'HTHHHH', 'THHHHH'}\n",
"assert update({'TTTTTTT'}, {3}) == {'HTTTTTT'}\n",
"assert (update(rotations('HHHHHT'), {0}) == update({'HHTHHH'}, {1}) == update({'THHHHH'}, {2})\n",
" == {'HHHHHH', 'HHHHTT', 'HHHTHT', 'HHTHHT'})\n",
"\n",
"assert coin_search(4) == [\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3}]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"How many distinct canonical coin sequences are there for up to a dozen coins?"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{1: 2,\n",
" 2: 3,\n",
" 3: 4,\n",
" 4: 6,\n",
" 5: 8,\n",
" 6: 14,\n",
" 7: 20,\n",
" 8: 36,\n",
" 9: 60,\n",
" 10: 108,\n",
" 11: 188,\n",
" 12: 352}"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{N: len(all_coins(N))\n",
" for N in range(1, 13)}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On the one hand this is encouraging; there are only 352 canonical coin sequences of length 10, far less than the 4,096 non-canonical squences. On the other hand, it is discouraging; since we are searching over belief states, that would be 2352 belief states, which is nore than a googol. However, we should be able to easily handle up to N=7, because 220 is only a million.\n",
"\n",
"# Winning Strategies for 1 to 7 Coins"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{1: [{0}],\n",
" 2: [{0, 1}, {0}, {0, 1}],\n",
" 3: None,\n",
" 4: [{0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3},\n",
" {0, 1},\n",
" {0, 1, 2, 3},\n",
" {0, 2},\n",
" {0, 1, 2, 3}],\n",
" 5: None,\n",
" 6: None,\n",
" 7: None}"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"{N: coin_search(N) for N in range(1, 8)}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Too bad; there are no winning strategies for N = 3, 5, 6, or 7. \n",
"\n",
"There *are* winning strategies for N = 1, 2, 4; they have lengths 1, 3, 15, respectively. Hmm. That suggests ...\n",
"\n",
"# A Conjecture\n",
"\n",
"> For every *N* that is a power of 2, there will be a shortest winning strategy of length 2*N* - 1.\n",
"\n",
"> For every *N* that is not a power of 2, there will be no winning strategy. \n",
"\n",
"# Winning Strategy for 8 Coins\n",
"\n",
"For N = 8, there are 236 = 69 billion belief states and if the conjecture is true there will be a shortest winning strategy with 255 steps. All the computations up to now have been instantaneous, but this one should take a few minutes. Let's see:"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 1min 18s, sys: 118 ms, total: 1min 18s\n",
"Wall time: 1min 18s\n"
]
}
],
"source": [
"%time strategy = coin_search(8)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"255"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"len(strategy)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Eureka!** That's evidence in favor of the conjecture. But not proof. And it leaves many questions unanswered:\n",
"- Can you show there are no winning strategies for *N* = 9, 10, 11, ...?\n",
"- Can you prove there are no winning strategies for any *N* that is not a power of 2?\n",
"- Can you find a winning strategy of length 65,535 for *N* = 16 and verify that it works?\n",
"- Can you generate a winning strategy for any power of 2 (without proving it is shortest)?\n",
"- Can you prove there are no shorter winning strategies for *N* = 16?\n",
"- Can you prove the conjecture in general?\n",
"- Can you *understand* and *explain* how the strategy works, rather than just listing the moves?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Visualizing Strategies\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def show(moves, N=4):\n",
" \"For each move, print the move number, move, and belief state.\"\n",
" belief = all_coins(N)\n",
" order = sorted(belief)\n",
" for (i, move) in enumerate(moves, 1):\n",
" belief = update(belief, move)\n",
" print('{:3} | {:8} | {}'.format(\n",
" i, movestr(move, N), beliefstr(belief, order)))\n",
"\n",
"def beliefstr(belief, order) -> str: \n",
" return join(((coins if coins in belief else ' ' * len(coins))\n",
" for coins in order), ' ')\n",
"\n",
"def movestr(move, N) -> str: \n",
" return join((i if i in move else ' ') \n",
" for i in range(N))\n",
" \n",
"def join(items, sep='') -> str: \n",
" return sep.join(map(str, items))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The following tables shows how moves change the belief state:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 1 | 01 | HH HT \n",
" 2 | 0 | HH TT\n",
" 3 | 01 | HH \n"
]
}
],
"source": [
"show(coin_search(2), 2)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 1 | 0123 | HHHH HHHT HHTT HTHT HTTT \n",
" 2 | 0 2 | HHHH HHHT HHTT HTTT TTTT\n",
" 3 | 0123 | HHHH HHHT HHTT HTTT \n",
" 4 | 01 | HHHH HHHT HTHT HTTT TTTT\n",
" 5 | 0123 | HHHH HHHT HTHT HTTT \n",
" 6 | 0 2 | HHHH HHHT HTTT TTTT\n",
" 7 | 0123 | HHHH HHHT HTTT \n",
" 8 | 012 | HHHH HHTT HTHT TTTT\n",
" 9 | 0123 | HHHH HHTT HTHT \n",
" 10 | 0 2 | HHHH HHTT TTTT\n",
" 11 | 0123 | HHHH HHTT \n",
" 12 | 01 | HHHH HTHT TTTT\n",
" 13 | 0123 | HHHH HTHT \n",
" 14 | 0 2 | HHHH TTTT\n",
" 15 | 0123 | HHHH \n"
]
}
],
"source": [
"show(coin_search(4))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see that every odd-numbered move flips all four coins to eliminate the possibility of `TTTT`, flipping it to `HHHH`. We can also see that moves 2, 4, and 6 flip two coins and have the effect of eventually eliminating the two \"two heads\" sequences from the belief state, and then move 8 eliminates the \"three heads\" and \"one heads\" sequences, while bringing back the \"two heads\" possibilities. Repeating moves 2, 4, and 6 in moves 10, 12, and 14 then re-eliminates the \"two heads\", and move 15 gets the belief state down to `{'HHHH'}`.\n",
"\n",
"You could call `show(strategy, 8)`, but the results look bad unless you have a very wide (345 characters) screen to view it on. So instead I'll add a `verbose` parameter to `play` and play out some games with a trace of each move:"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 1: HHTH rot: HHHT flip: 0123 => HTTT\n",
" 2: HTTT rot: TTTH flip: 0 2 => HHHT\n",
" 3: HHHT rot: HTHH flip: 0123 => HTTT\n",
" 4: HTTT rot: HTTT flip: 01 => HTTT\n",
" 5: HTTT rot: TTHT flip: 0123 => HHHT\n",
" 6: HHHT rot: THHH flip: 0 2 => HHHT\n",
" 7: HHHT rot: THHH flip: 0123 => HTTT\n",
" 8: HTTT rot: TTTH flip: 012 => HHHH\n"
]
},
{
"data": {
"text/plain": [
"'HHHH'"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def play(coins, strategy, verbose=False):\n",
" \"Play strategy against a random Devil; return final state of coins.\"\n",
" N = len(coins)\n",
" for i, move in enumerate(strategy, 1):\n",
" if 'T' not in coins: return coins\n",
" coins0 = coins\n",
" coins1 = random.choice(list(rotations(coins)))\n",
" coins = flip(coins1, move)\n",
" if verbose: \n",
" print('{:4d}: {} rot: {} flip: {} => {}'.format(\n",
" i, coins0, coins1, movestr(move, N), coins))\n",
" return coins\n",
"\n",
"play('HHTH', coin_search(4), True)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 1: HTTHTTHT rot: TTHTHTTH flip: 01234567 => HHTHHTHT\n",
" 2: HHTHHTHT rot: HHTHTHHT flip: 0 2 4 6 => HHHHHTTT\n",
" 3: HHHHHTTT rot: TTHHHHHT flip: 01234567 => HHHTTTTT\n",
" 4: HHHTTTTT rot: HTTTTTHH flip: 01 45 => HHHHTHTT\n",
" 5: HHHHTHTT rot: HTHTTHHH flip: 01234567 => HHTTTTHT\n",
" 6: HHTTTTHT rot: TTTHTHHT flip: 0 2 4 6 => HHHHTTHT\n",
" 7: HHHHTTHT rot: TTHTHHHH flip: 01234567 => HHTHTTTT\n",
" 8: HHTHTTTT rot: THHTHTTT flip: 012 456 => HHTHTTTT\n",
" 9: HHTHTTTT rot: THHTHTTT flip: 01234567 => HHHHTTHT\n",
" 10: HHHHTTHT rot: TTHTHHHH flip: 0 2 4 6 => HHTTTTHT\n",
" 11: HHTTTTHT rot: HTHHTTTT flip: 01234567 => HHHHTHTT\n",
" 12: HHHHTHTT rot: HTTHHHHT flip: 01 45 => HTHTTHTT\n",
" 13: HTHTTHTT rot: THTTHTTH flip: 01234567 => HHTHHTHT\n",
" 14: HHTHHTHT rot: THHTHTHH flip: 0 2 4 6 => HHHTTTTT\n",
" 15: HHHTTTTT rot: TTTTTHHH flip: 01234567 => HHHHHTTT\n",
" 16: HHHHHTTT rot: HHHTTTHH flip: 0123 => HHTTTHTT\n",
" 17: HHTTTHTT rot: HHTTTHTT flip: 01234567 => HHHTHHTT\n",
" 18: HHHTHHTT rot: HHHTHHTT flip: 0 2 4 6 => HHTTHTTT\n",
" 19: HHTTHTTT rot: TTHTTTHH flip: 01234567 => HHHTTHHT\n",
" 20: HHHTTHHT rot: HHHTTHHT flip: 01 45 => HTHTHTTT\n",
" 21: HTHTHTTT rot: TTHTHTHT flip: 01234567 => HHHTHTHT\n",
" 22: HHHTHTHT rot: HHHTHTHT flip: 0 2 4 6 => HTTTTTTT\n",
" 23: HTTTTTTT rot: TTTTTTTH flip: 01234567 => HHHHHHHT\n",
" 24: HHHHHHHT rot: HHHHHHHT flip: 012 456 => HTTTTTTT\n",
" 25: HTTTTTTT rot: TTTTTHTT flip: 01234567 => HHHHHHHT\n",
" 26: HHHHHHHT rot: HHHHHHHT flip: 0 2 4 6 => HTHTHTTT\n",
" 27: HTHTHTTT rot: THTTTHTH flip: 01234567 => HHHTHTHT\n",
" 28: HHHTHTHT rot: HHHTHTHT flip: 01 45 => HHTTTHTT\n",
" 29: HHTTTHTT rot: TTHHTTTH flip: 01234567 => HHHTHHTT\n",
" 30: HHHTHHTT rot: TTHHHTHH flip: 0 2 4 6 => HHTTHTTT\n",
" 31: HHTTHTTT rot: HTTHTTTH flip: 01234567 => HHHTTHHT\n",
" 32: HHHTTHHT rot: TTHHTHHH flip: 01234 6 => HHHTTHHT\n",
" 33: HHHTTHHT rot: THHHTTHH flip: 01234567 => HHTTHTTT\n",
" 34: HHTTHTTT rot: TTTHHTTH flip: 0 2 4 6 => HHHTHHTT\n",
" 35: HHHTHHTT rot: HTTHHHTH flip: 01234567 => HHTTTHTT\n",
" 36: HHTTTHTT rot: TTHHTTTH flip: 01 45 => HHHHHHHT\n",
" 37: HHHHHHHT rot: HTHHHHHH flip: 01234567 => HTTTTTTT\n",
" 38: HTTTTTTT rot: HTTTTTTT flip: 0 2 4 6 => HTHTHTTT\n",
" 39: HTHTHTTT rot: THTHTTTH flip: 01234567 => HHHTHTHT\n",
" 40: HHHTHTHT rot: HHHTHTHT flip: 012 456 => HTTTTTTT\n",
" 41: HTTTTTTT rot: TTTTTTTH flip: 01234567 => HHHHHHHT\n",
" 42: HHHHHHHT rot: HHTHHHHH flip: 0 2 4 6 => HHHTHTHT\n",
" 43: HHHTHTHT rot: HTHHHTHT flip: 01234567 => HTHTHTTT\n",
" 44: HTHTHTTT rot: HTHTHTTT flip: 01 45 => HHTTHTTT\n",
" 45: HHTTHTTT rot: TTHTTTHH flip: 01234567 => HHHTTHHT\n",
" 46: HHHTTHHT rot: HTTHHTHH flip: 0 2 4 6 => HHTTTHTT\n",
" 47: HHTTTHTT rot: THTTHHTT flip: 01234567 => HHHTHHTT\n",
" 48: HHHTHHTT rot: HHTHHTTH flip: 0123 => HTHTTHTT\n",
" 49: HTHTTHTT rot: TTHTTHTH flip: 01234567 => HHTHHTHT\n",
" 50: HHTHHTHT rot: THTHHTHH flip: 0 2 4 6 => HHHHHTTT\n",
" 51: HHHHHTTT rot: HTTTHHHH flip: 01234567 => HHHTTTTT\n",
" 52: HHHTTTTT rot: TTHHHTTT flip: 01 45 => HHHHTHTT\n",
" 53: HHHHTHTT rot: HHHHTHTT flip: 01234567 => HHTTTTHT\n",
" 54: HHTTTTHT rot: TTTTHTHH flip: 0 2 4 6 => HHTHTTTT\n",
" 55: HHTHTTTT rot: THTTTTHH flip: 01234567 => HHHHTTHT\n",
" 56: HHHHTTHT rot: HHTTHTHH flip: 012 456 => HTHTTHTT\n",
" 57: HTHTTHTT rot: THTTHTTH flip: 01234567 => HHTHHTHT\n",
" 58: HHTHHTHT rot: HHTHTHHT flip: 0 2 4 6 => HHHHHTTT\n",
" 59: HHHHHTTT rot: HHTTTHHH flip: 01234567 => HHHTTTTT\n",
" 60: HHHTTTTT rot: TTTTHHHT flip: 01 45 => HHTTTTHT\n",
" 61: HHTTTTHT rot: HTTTTHTH flip: 01234567 => HHHHTHTT\n",
" 62: HHHHTHTT rot: THHHHTHT flip: 0 2 4 6 => HHTHTTTT\n",
" 63: HHTHTTTT rot: THTTTTHH flip: 01234567 => HHHHTTHT\n",
" 64: HHHHTTHT rot: HHHTTHTH flip: 012345 => HHTTHTTT\n",
" 65: HHTTHTTT rot: HTTHTTTH flip: 01234567 => HHHTTHHT\n",
" 66: HHHTTHHT rot: TTHHTHHH flip: 0 2 4 6 => HHHTHHTT\n",
" 67: HHHTHHTT rot: HHTHHTTH flip: 01234567 => HHTTTHTT\n",
" 68: HHTTTHTT rot: THTTHHTT flip: 01 45 => HTTTTTTT\n",
" 69: HTTTTTTT rot: TTTTTTHT flip: 01234567 => HHHHHHHT\n",
" 70: HHHHHHHT rot: HHHTHHHH flip: 0 2 4 6 => HTHTHTTT\n",
" 71: HTHTHTTT rot: TTHTHTHT flip: 01234567 => HHHTHTHT\n",
" 72: HHHTHTHT rot: HTHHHTHT flip: 012 456 => HTHTHTTT\n",
" 73: HTHTHTTT rot: TTHTHTHT flip: 01234567 => HHHTHTHT\n",
" 74: HHHTHTHT rot: HTHTHTHH flip: 0 2 4 6 => HTTTTTTT\n",
" 75: HTTTTTTT rot: TTTTHTTT flip: 01234567 => HHHHHHHT\n",
" 76: HHHHHHHT rot: HHTHHHHH flip: 01 45 => HHTTTHTT\n",
" 77: HHTTTHTT rot: HTTTHTTH flip: 01234567 => HHHTHHTT\n",
" 78: HHHTHHTT rot: HHTTHHHT flip: 0 2 4 6 => HHTTHTTT\n",
" 79: HHTTHTTT rot: HHTTHTTT flip: 01234567 => HHHTTHHT\n",
" 80: HHHTTHHT rot: HTHHHTTH flip: 0123 => HTHTTHTT\n",
" 81: HTHTTHTT rot: THTTHTTH flip: 01234567 => HHTHHTHT\n",
" 82: HHTHHTHT rot: HHTHHTHT flip: 0 2 4 6 => HHHTTTTT\n",
" 83: HHHTTTTT rot: TTHHHTTT flip: 01234567 => HHHHHTTT\n",
" 84: HHHHHTTT rot: TTHHHHHT flip: 01 45 => HHHHTTHT\n",
" 85: HHHHTTHT rot: HTTHTHHH flip: 01234567 => HHTHTTTT\n",
" 86: HHTHTTTT rot: HTTTTHHT flip: 0 2 4 6 => HHTTTTHT\n",
" 87: HHTTTTHT rot: TTTTHTHH flip: 01234567 => HHHHTHTT\n",
" 88: HHHHTHTT rot: THHHHTHT flip: 012 456 => HTHTTHTT\n",
" 89: HTHTTHTT rot: THTTHTTH flip: 01234567 => HHTHHTHT\n",
" 90: HHTHHTHT rot: THTHHTHH flip: 0 2 4 6 => HHHHHTTT\n",
" 91: HHHHHTTT rot: TTTHHHHH flip: 01234567 => HHHTTTTT\n",
" 92: HHHTTTTT rot: HHHTTTTT flip: 01 45 => HHTTTTHT\n",
" 93: HHTTTTHT rot: TTHTHHTT flip: 01234567 => HHHHTHTT\n",
" 94: HHHHTHTT rot: HHTHTTHH flip: 0 2 4 6 => HHHHTTHT\n",
" 95: HHHHTTHT rot: TTHTHHHH flip: 01234567 => HHTHTTTT\n",
" 96: HHTHTTTT rot: TTHHTHTT flip: 01234 6 => HHHTHHTT\n",
" 97: HHHTHHTT rot: TTHHHTHH flip: 01234567 => HHTTTHTT\n",
" 98: HHTTTHTT rot: TTHTTHHT flip: 0 2 4 6 => HHTTHTTT\n",
" 99: HHTTHTTT rot: THTTTHHT flip: 01234567 => HHHTTHHT\n",
" 100: HHHTTHHT rot: THHTHHHT flip: 01 45 => HTHTHTTT\n",
" 101: HTHTHTTT rot: THTHTTTH flip: 01234567 => HHHTHTHT\n",
" 102: HHHTHTHT rot: HTHTHTHH flip: 0 2 4 6 => HTTTTTTT\n",
" 103: HTTTTTTT rot: TTTTTHTT flip: 01234567 => HHHHHHHT\n",
" 104: HHHHHHHT rot: HHHHHHHT flip: 012 456 => HTTTTTTT\n",
" 105: HTTTTTTT rot: TTTHTTTT flip: 01234567 => HHHHHHHT\n",
" 106: HHHHHHHT rot: HHHHHHHT flip: 0 2 4 6 => HTHTHTTT\n",
" 107: HTHTHTTT rot: THTHTHTT flip: 01234567 => HHHTHTHT\n",
" 108: HHHTHTHT rot: HTHTHTHH flip: 01 45 => HHHTHHTT\n",
" 109: HHHTHHTT rot: TTHHHTHH flip: 01234567 => HHTTTHTT\n",
" 110: HHTTTHTT rot: HTTTHTTH flip: 0 2 4 6 => HHTTHTTT\n",
" 111: HHTTHTTT rot: TTTHHTTH flip: 01234567 => HHHTTHHT\n",
" 112: HHHTTHHT rot: HHHTTHHT flip: 0123 => HHTTTTHT\n",
" 113: HHTTTTHT rot: TTHTHHTT flip: 01234567 => HHHHTHTT\n",
" 114: HHHHTHTT rot: HHTHTTHH flip: 0 2 4 6 => HHHHTTHT\n",
" 115: HHHHTTHT rot: HHHHTTHT flip: 01234567 => HHTHTTTT\n",
" 116: HHTHTTTT rot: HTTTTHHT flip: 01 45 => HTHTTHTT\n",
" 117: HTHTTHTT rot: THTHTTHT flip: 01234567 => HHTHHTHT\n",
" 118: HHTHHTHT rot: HHTHHTHT flip: 0 2 4 6 => HHHTTTTT\n",
" 119: HHHTTTTT rot: HHTTTTTH flip: 01234567 => HHHHHTTT\n",
" 120: HHHHHTTT rot: TTTHHHHH flip: 012 456 => HHHHHTTT\n",
" 121: HHHHHTTT rot: HTTTHHHH flip: 01234567 => HHHTTTTT\n",
" 122: HHHTTTTT rot: HHHTTTTT flip: 0 2 4 6 => HTHTTHTT\n",
" 123: HTHTTHTT rot: THTTHTHT flip: 01234567 => HHTHHTHT\n",
" 124: HHTHHTHT rot: THHTHHTH flip: 01 45 => HHTHTTTT\n",
" 125: HHTHTTTT rot: HHTHTTTT flip: 01234567 => HHHHTTHT\n",
" 126: HHHHTTHT rot: THHHHTTH flip: 0 2 4 6 => HHHHTHTT\n",
" 127: HHHHTHTT rot: THHHHTHT flip: 01234567 => HHTTTTHT\n",
" 128: HHTTTTHT rot: HTTTTHTH flip: 0123456 => HHHHTHHT\n",
" 129: HHHHTHHT rot: THHTHHHH flip: 01234567 => HTTHTTTT\n",
" 130: HTTHTTTT rot: TTHTTTTH flip: 0 2 4 6 => HHHTTTHT\n",
" 131: HHHTTTHT rot: THHHTTTH flip: 01234567 => HHHTHTTT\n",
" 132: HHHTHTTT rot: THHHTHTT flip: 01 45 => HHHTTTHT\n",
" 133: HHHTTTHT rot: TTTHTHHH flip: 01234567 => HHHTHTTT\n",
" 134: HHHTHTTT rot: HTTTHHHT flip: 0 2 4 6 => HTTHTTTT\n",
" 135: HTTHTTTT rot: TTTTHTTH flip: 01234567 => HHHHTHHT\n",
" 136: HHHHTHHT rot: HHHTHHTH flip: 012 456 => HHTTTTTT\n",
" 137: HHTTTTTT rot: TTHHTTTT flip: 01234567 => HHHHHHTT\n",
" 138: HHHHHHTT rot: HHTTHHHH flip: 0 2 4 6 => HHTTHTHT\n",
" 139: HHTTHTHT rot: HTHTHHTT flip: 01234567 => HHTHTHTT\n",
" 140: HHTHTHTT rot: THTTHHTH flip: 01 45 => HHTTTTTT\n",
" 141: HHTTTTTT rot: TTTHHTTT flip: 01234567 => HHHHHHTT\n",
" 142: HHHHHHTT rot: TTHHHHHH flip: 0 2 4 6 => HHTTHTHT\n",
" 143: HHTTHTHT rot: TTHTHTHH flip: 01234567 => HHTHTHTT\n",
" 144: HHTHTHTT rot: HTTHHTHT flip: 0123 => HHTHTHTT\n",
" 145: HHTHTHTT rot: TTHHTHTH flip: 01234567 => HHTTHTHT\n",
" 146: HHTTHTHT rot: HHTTHTHT flip: 0 2 4 6 => HHTTTTTT\n",
" 147: HHTTTTTT rot: TTHHTTTT flip: 01234567 => HHHHHHTT\n",
" 148: HHHHHHTT rot: HHHHHHTT flip: 01 45 => HHTTTTTT\n",
" 149: HHTTTTTT rot: TTTTHHTT flip: 01234567 => HHHHHHTT\n",
" 150: HHHHHHTT rot: HHHHTTHH flip: 0 2 4 6 => HHTTHTHT\n",
" 151: HHTTHTHT rot: THTHHTTH flip: 01234567 => HHTHTHTT\n",
" 152: HHTHTHTT rot: HHTHTHTT flip: 012 456 => HHHTHTTT\n",
" 153: HHHTHTTT rot: THTTTHHH flip: 01234567 => HHHTTTHT\n",
" 154: HHHTTTHT rot: HHTTTHTH flip: 0 2 4 6 => HHHHTHHT\n",
" 155: HHHHTHHT rot: THHTHHHH flip: 01234567 => HTTHTTTT\n",
" 156: HTTHTTTT rot: TTHTTTTH flip: 01 45 => HHHHTHHT\n",
" 157: HHHHTHHT rot: HHTHHHHT flip: 01234567 => HTTHTTTT\n",
" 158: HTTHTTTT rot: TTTHTTHT flip: 0 2 4 6 => HHHTTTHT\n",
" 159: HHHTTTHT rot: HHHTTTHT flip: 01234567 => HHHTHTTT\n",
" 160: HHHTHTTT rot: TTHHHTHT flip: 01234 6 => HHTTTTTT\n",
" 161: HHTTTTTT rot: TTHHTTTT flip: 01234567 => HHHHHHTT\n",
" 162: HHHHHHTT rot: HHHHHTTH flip: 0 2 4 6 => HHTHTHTT\n",
" 163: HHTHTHTT rot: THTTHHTH flip: 01234567 => HHTTHTHT\n",
" 164: HHTTHTHT rot: TTHTHTHH flip: 01 45 => HHHHHHTT\n",
" 165: HHHHHHTT rot: HHTTHHHH flip: 01234567 => HHTTTTTT\n",
" 166: HHTTTTTT rot: HHTTTTTT flip: 0 2 4 6 => HHTHTHTT\n",
" 167: HHTHTHTT rot: THTHTTHH flip: 01234567 => HHTTHTHT\n",
" 168: HHTTHTHT rot: HTHHTTHT flip: 012 456 => HHHTTTHT\n",
" 169: HHHTTTHT rot: TTTHTHHH flip: 01234567 => HHHTHTTT\n",
" 170: HHHTHTTT rot: TTHHHTHT flip: 0 2 4 6 => HTTHTTTT\n",
" 171: HTTHTTTT rot: TTTTHTTH flip: 01234567 => HHHHTHHT\n",
" 172: HHHHTHHT rot: HTHHTHHH flip: 01 45 => HHHHTHHT\n",
" 173: HHHHTHHT rot: THHTHHHH flip: 01234567 => HTTHTTTT\n",
" 174: HTTHTTTT rot: TTHTTHTT flip: 0 2 4 6 => HHHTHTTT\n",
" 175: HHHTHTTT rot: THTTTHHH flip: 01234567 => HHHTTTHT\n",
" 176: HHHTTTHT rot: HHHTTTHT flip: 0123 => HTTHTTTT\n",
" 177: HTTHTTTT rot: TTHTTTTH flip: 01234567 => HHHHTHHT\n",
" 178: HHHHTHHT rot: HHTHHTHH flip: 0 2 4 6 => HHHTTTHT\n",
" 179: HHHTTTHT rot: HHHTTTHT flip: 01234567 => HHHTHTTT\n",
" 180: HHHTHTTT rot: TTTHHHTH flip: 01 45 => HHHTHTTT\n",
" 181: HHHTHTTT rot: HTTTHHHT flip: 01234567 => HHHTTTHT\n",
" 182: HHHTTTHT rot: HHTTTHTH flip: 0 2 4 6 => HHHHTHHT\n",
" 183: HHHHTHHT rot: THHHHTHH flip: 01234567 => HTTHTTTT\n",
" 184: HTTHTTTT rot: HTTTTHTT flip: 012 456 => HHTHTHTT\n",
" 185: HHTHTHTT rot: THHTHTHT flip: 01234567 => HHTTHTHT\n",
" 186: HHTTHTHT rot: HTTHTHTH flip: 0 2 4 6 => HHHHHHTT\n",
" 187: HHHHHHTT rot: TTHHHHHH flip: 01234567 => HHTTTTTT\n",
" 188: HHTTTTTT rot: TTTTTTHH flip: 01 45 => HHHHHHTT\n",
" 189: HHHHHHTT rot: HHHHHTTH flip: 01234567 => HHTTTTTT\n",
" 190: HHTTTTTT rot: HHTTTTTT flip: 0 2 4 6 => HHTHTHTT\n",
" 191: HHTHTHTT rot: HTHTTHHT flip: 01234567 => HHTTHTHT\n",
" 192: HHTTHTHT rot: HTHTHHTT flip: 012345 => HTHTTTTT\n",
" 193: HTHTTTTT rot: HTHTTTTT flip: 01234567 => HHHHHTHT\n",
" 194: HHHHHTHT rot: THHHHHTH flip: 0 2 4 6 => HHHHHTHT\n",
" 195: HHHHHTHT rot: HTHTHHHH flip: 01234567 => HTHTTTTT\n",
" 196: HTHTTTTT rot: TTTTHTHT flip: 01 45 => HHTHHTTT\n",
" 197: HHTHHTTT rot: HTHHTTTH flip: 01234567 => HHHTTHTT\n",
" 198: HHHTTHTT rot: HTTHHHTT flip: 0 2 4 6 => HHTHHTTT\n",
" 199: HHTHHTTT rot: THHTHHTT flip: 01234567 => HHHTTHTT\n",
" 200: HHHTTHTT rot: HTTHHHTT flip: 012 456 => HHHTTHTT\n",
" 201: HHHTTHTT rot: HHHTTHTT flip: 01234567 => HHTHHTTT\n",
" 202: HHTHHTTT rot: HTHHTTTH flip: 0 2 4 6 => HHTHHTTT\n",
" 203: HHTHHTTT rot: TTTHHTHH flip: 01234567 => HHHTTHTT\n",
" 204: HHHTTHTT rot: TTHTTHHH flip: 01 45 => HHHHHTHT\n",
" 205: HHHHHTHT rot: HHHHHTHT flip: 01234567 => HTHTTTTT\n",
" 206: HTHTTTTT rot: TTHTHTTT flip: 0 2 4 6 => HTHTTTTT\n",
" 207: HTHTTTTT rot: TTTHTHTT flip: 01234567 => HHHHHTHT\n",
" 208: HHHHHTHT rot: THHHHHTH flip: 0123 => HHTHHTTT\n",
" 209: HHTHHTTT rot: HTTTHHTH flip: 01234567 => HHHTTHTT\n",
" 210: HHHTTHTT rot: TTHHHTTH flip: 0 2 4 6 => HHHTTHTT\n",
" 211: HHHTTHTT rot: HTTHHHTT flip: 01234567 => HHTHHTTT\n",
" 212: HHTHHTTT rot: TTTHHTHH flip: 01 45 => HHHHHTHT\n",
" 213: HHHHHTHT rot: HHHHHTHT flip: 01234567 => HTHTTTTT\n",
" 214: HTHTTTTT rot: THTHTTTT flip: 0 2 4 6 => HHHHHTHT\n",
" 215: HHHHHTHT rot: THTHHHHH flip: 01234567 => HTHTTTTT\n",
" 216: HTHTTTTT rot: HTTTTTHT flip: 012 456 => HHTHHTTT\n",
" 217: HHTHHTTT rot: THHTTTHH flip: 01234567 => HHHTTHTT\n",
" 218: HHHTTHTT rot: THTTHHHT flip: 0 2 4 6 => HHHTTHTT\n",
" 219: HHHTTHTT rot: TTHTTHHH flip: 01234567 => HHTHHTTT\n",
" 220: HHTHHTTT rot: THHTHHTT flip: 01 45 => HTHTTTTT\n",
" 221: HTHTTTTT rot: TTHTHTTT flip: 01234567 => HHHHHTHT\n",
" 222: HHHHHTHT rot: HHHTHTHH flip: 0 2 4 6 => HTHTTTTT\n",
" 223: HTHTTTTT rot: TTTTHTHT flip: 01234567 => HHHHHTHT\n",
" 224: HHHHHTHT rot: THTHHHHH flip: 01234 6 => HHTHTTHT\n",
" 225: HHTHTTHT rot: HTHTTHTH flip: 01234567 => HHTHTTHT\n",
" 226: HHTHTTHT rot: HTTHTHHT flip: 0 2 4 6 => HHHHTTTT\n",
" 227: HHHHTTTT rot: HHTTTTHH flip: 01234567 => HHHHTTTT\n",
" 228: HHHHTTTT rot: TTHHHHTT flip: 01 45 => HHHHTTTT\n",
" 229: HHHHTTTT rot: TTHHHHTT flip: 01234567 => HHHHTTTT\n",
" 230: HHHHTTTT rot: TTHHHHTT flip: 0 2 4 6 => HHTHTTHT\n",
" 231: HHTHTTHT rot: HHTHTTHT flip: 01234567 => HHTHTTHT\n",
" 232: HHTHTTHT rot: HHTHTTHT flip: 012 456 => HHHHTTTT\n",
" 233: HHHHTTTT rot: TTHHHHTT flip: 01234567 => HHHHTTTT\n",
" 234: HHHHTTTT rot: TTHHHHTT flip: 0 2 4 6 => HHTHTTHT\n",
" 235: HHTHTTHT rot: TTHTHHTH flip: 01234567 => HHTHTTHT\n",
" 236: HHTHTTHT rot: THTHHTHT flip: 01 45 => HHTHTTHT\n",
" 237: HHTHTTHT rot: THTHHTHT flip: 01234567 => HHTHTTHT\n",
" 238: HHTHTTHT rot: TTHTHHTH flip: 0 2 4 6 => HHHHTTTT\n",
" 239: HHHHTTTT rot: TTTHHHHT flip: 01234567 => HHHHTTTT\n",
" 240: HHHHTTTT rot: TTTHHHHT flip: 0123 => HHHTHHHT\n",
" 241: HHHTHHHT rot: HHHTHHHT flip: 01234567 => HTTTHTTT\n",
" 242: HTTTHTTT rot: TTHTTTHT flip: 0 2 4 6 => HTTTHTTT\n",
" 243: HTTTHTTT rot: HTTTHTTT flip: 01234567 => HHHTHHHT\n",
" 244: HHHTHHHT rot: HHTHHHTH flip: 01 45 => HTTTHTTT\n",
" 245: HTTTHTTT rot: TTHTTTHT flip: 01234567 => HHHTHHHT\n",
" 246: HHHTHHHT rot: HHHTHHHT flip: 0 2 4 6 => HTTTHTTT\n",
" 247: HTTTHTTT rot: THTTTHTT flip: 01234567 => HHHTHHHT\n",
" 248: HHHTHHHT rot: HHTHHHTH flip: 012 456 => HHTTHHTT\n",
" 249: HHTTHHTT rot: HHTTHHTT flip: 01234567 => HHTTHHTT\n",
" 250: HHTTHHTT rot: HHTTHHTT flip: 0 2 4 6 => HHTTHHTT\n",
" 251: HHTTHHTT rot: HHTTHHTT flip: 01234567 => HHTTHHTT\n",
" 252: HHTTHHTT rot: THHTTHHT flip: 01 45 => HTHTHTHT\n",
" 253: HTHTHTHT rot: HTHTHTHT flip: 01234567 => HTHTHTHT\n",
" 254: HTHTHTHT rot: HTHTHTHT flip: 0 2 4 6 => TTTTTTTT\n",
" 255: TTTTTTTT rot: TTTTTTTT flip: 01234567 => HHHHHHHH\n"
]
},
{
"data": {
"text/plain": [
"'HHHHHHHH'"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"play('HTTHTTHT', strategy, True)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.5.3"
}
},
"nbformat": 4,
"nbformat_minor": 2
}