diff --git a/ipynb/Coin Flip.ipynb b/ipynb/Coin Flip.ipynb index 000af0e..4409ee5 100644 --- a/ipynb/Coin Flip.ipynb +++ b/ipynb/Coin Flip.ipynb @@ -4,6 +4,8 @@ "cell_type": "markdown", "metadata": {}, "source": [ + "
Peter Norvig
\n", + "\n", "# 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", @@ -16,6 +18,8 @@ "\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", + "(This same puzzle also appeared in [The Riddler on 21 June 2019](https://fivethirtyeight.com/features/i-would-walk-500-miles-and-i-would-riddle-500-more/), with the role of the devil played by a banker. I'm not sure which is scarier.)\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", @@ -76,9 +80,7 @@ { "cell_type": "code", "execution_count": 2, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "def all_moves() -> [Move]: \n", @@ -237,7 +239,7 @@ "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:" + "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 (because if there had been 4 heads, you would have already won), leaving 15 possible coin sequences:" ] }, { @@ -315,9 +317,7 @@ { "cell_type": "code", "execution_count": 9, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "def search(start, goal, actions, result) -> Strategy:\n", @@ -444,7 +444,7 @@ "\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)." + "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)." ] }, { @@ -461,9 +461,7 @@ { "cell_type": "code", "execution_count": 13, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "assert Coins('HHHT') == Coins('HHTH') == Coins('HTHH') == Coins('THHH') == 'HHHT'" @@ -502,7 +500,7 @@ "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:" + "Let's make sure we didn't break anything and that we get the same 15-step solution:" ] }, { @@ -553,9 +551,7 @@ { "cell_type": "code", "execution_count": 16, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "def all_moves(N=4) -> [Move]:\n", @@ -580,15 +576,13 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Let's test the new definitions and make sure we haven't broken `update` and `coin_search`:" + "Let's test the new definitions:" ] }, { "cell_type": "code", "execution_count": 17, - "metadata": { - "collapsed": true - }, + "metadata": {}, "outputs": [], "source": [ "assert all_moves(3) == [{0, 1, 2}, {0, 1}, {0}, set()]\n", @@ -663,7 +657,7 @@ "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", + "On the one hand this is encouraging; there are only 352 canonical coin sequences of length 12, 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" ] @@ -718,13 +712,12 @@ "\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", + "- For every N that is a power of 2, there will be a shortest winning strategy of length 2N - 1.\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:" + "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 less than a second, but this one should take more than a minute. Let's see:" ] }, { @@ -736,8 +729,8 @@ "name": "stdout", "output_type": "stream", "text": [ - "CPU times: user 1min 18s, sys: 118 ms, total: 1min 18s\n", - "Wall time: 1min 18s\n" + "CPU times: user 1min 10s, sys: 119 ms, total: 1min 11s\n", + "Wall time: 1min 11s\n" ] } ], @@ -784,25 +777,27 @@ "metadata": {}, "source": [ "# Visualizing Strategies\n", + "\n", + "Can I understand what is going on by showing how belief states are whittled down?\n", "\n" ] }, { "cell_type": "code", "execution_count": 22, - "metadata": { - "collapsed": true - }, + "metadata": {}, "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", + " def show(i, move, order=sorted(belief)):\n", + " print('{:3} | {:8} | {}'.format(i, movestr(move, N), beliefstr(belief, order)))\n", + " show(0, set())\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", + " show(i, move)\n", "\n", "def beliefstr(belief, order) -> str: \n", " return join(((coins if coins in belief else ' ' * len(coins))\n", @@ -832,6 +827,7 @@ "name": "stdout", "output_type": "stream", "text": [ + " 0 | | HH HT TT\n", " 1 | 01 | HH HT \n", " 2 | 0 | HH TT\n", " 3 | 01 | HH \n" @@ -851,6 +847,7 @@ "name": "stdout", "output_type": "stream", "text": [ + " 0 | | HHHH HHHT HHTT HTHT HTTT TTTT\n", " 1 | 0123 | HHHH HHHT HHTT HTHT HTTT \n", " 2 | 0 2 | HHHH HHHT HHTT HTTT TTTT\n", " 3 | 0123 | HHHH HHHT HHTT HTTT \n", @@ -891,14 +888,15 @@ "name": "stdout", "output_type": "stream", "text": [ - " 1: HHTH rot: HHHT flip: 0123 => HTTT\n", + " 1: HHTH rot: HTHH 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", + " 3: HHHT rot: HHTH flip: 0123 => HTTT\n", + " 4: HTTT rot: THTT 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" + " 6: HHHT rot: HTHH flip: 0 2 => HTTT\n", + " 7: HTTT rot: HTTT flip: 0123 => HHHT\n", + " 8: HHHT rot: HHHT flip: 012 => TTTT\n", + " 9: TTTT rot: TTTT flip: 0123 => HHHH\n" ] }, { @@ -938,261 +936,158 @@ "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", + " 1: HTTHTTHT rot: HTTHTHTT flip: 01234567 => HHTHHTHT\n", + " 2: HHTHHTHT rot: HTHHTHHT flip: 0 2 4 6 => HHHTTTTT\n", + " 3: HHHTTTTT rot: TTHHHTTT flip: 01234567 => HHHHHTTT\n", + " 4: HHHHHTTT rot: THHHHHTT flip: 01 45 => HHTTTTHT\n", + " 5: HHTTTTHT rot: THTHHTTT flip: 01234567 => HHHHTHTT\n", + " 6: HHHHTHTT rot: HHTHTTHH flip: 0 2 4 6 => HHHHTTHT\n", + " 7: HHHHTTHT rot: HHHTTHTH flip: 01234567 => HHTHTTTT\n", + " 8: HHTHTTTT rot: HTTTTHHT 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", + " 10: HHHHTTHT rot: HTHHHHTT flip: 0 2 4 6 => HHTTTTHT\n", + " 11: HHTTTTHT rot: THHTTTTH flip: 01234567 => HHHHTHTT\n", + " 12: HHHHTHTT rot: THHHHTHT flip: 01 45 => HHTHHTHT\n", + " 13: HHTHHTHT rot: THHTHTHH flip: 01234567 => HTHTTHTT\n", + " 14: HTHTTHTT rot: THTTHTHT flip: 0 2 4 6 => HHHTTTTT\n", + " 15: HHHTTTTT rot: TTTHHHTT flip: 01234567 => HHHHHTTT\n", + " 16: HHHHHTTT rot: THHHHHTT flip: 0123 => HHTTHTTT\n", + " 17: HHTTHTTT rot: TTHHTTHT flip: 01234567 => HHHTTHHT\n", + " 18: HHHTTHHT rot: THHTHHHT flip: 0 2 4 6 => HHTTTHTT\n", + " 19: HHTTTHTT rot: TTHTTHHT flip: 01234567 => HHHTHHTT\n", + " 20: HHHTHHTT rot: THHTTHHH flip: 01 45 => HHHTHTHT\n", + " 21: HHHTHTHT rot: THTHHHTH flip: 01234567 => HTHTHTTT\n", + " 22: HTHTHTTT rot: THTTTHTH flip: 0 2 4 6 => HHHHHHHT\n", + " 23: HHHHHHHT rot: THHHHHHH flip: 01234567 => HTTTTTTT\n", + " 24: HTTTTTTT rot: TTTTTTHT flip: 012 456 => HHHTHHTT\n", + " 25: HHHTHHTT rot: HHTTHHHT flip: 01234567 => HHTTTHTT\n", + " 26: HHTTTHTT rot: TTHHTTTH flip: 0 2 4 6 => HHHTTHHT\n", + " 27: HHHTTHHT rot: HTTHHTHH flip: 01234567 => HHTTHTTT\n", + " 28: HHTTHTTT rot: TTTHHTTH flip: 01 45 => HHHTHTHT\n", + " 29: HHHTHTHT rot: THTHHHTH flip: 01234567 => HTHTHTTT\n", + " 30: HTHTHTTT rot: THTTTHTH flip: 0 2 4 6 => HHHHHHHT\n", + " 31: HHHHHHHT rot: HHHHTHHH flip: 01234567 => HTTTTTTT\n", + " 32: HTTTTTTT rot: THTTTTTT flip: 01234 6 => HHHTHTHT\n", + " 33: HHHTHTHT rot: HTHHHTHT flip: 01234567 => HTHTHTTT\n", + " 34: HTHTHTTT rot: TTHTHTHT flip: 0 2 4 6 => HTTTTTTT\n", + " 35: HTTTTTTT rot: TTTTTHTT flip: 01234567 => HHHHHHHT\n", + " 36: HHHHHHHT rot: HTHHHHHH flip: 01 45 => HHHTTHHT\n", + " 37: HHHTTHHT rot: HHTTHHTH flip: 01234567 => HHTTHTTT\n", + " 38: HHTTHTTT rot: HHTTHTTT flip: 0 2 4 6 => HHTTTHTT\n", + " 39: HHTTTHTT rot: TTHTTHHT flip: 01234567 => HHHTHHTT\n", + " 40: HHHTHHTT rot: HHHTHHTT flip: 012 456 => HTTTTTTT\n", + " 41: HTTTTTTT rot: THTTTTTT flip: 01234567 => HHHHHHHT\n", + " 42: HHHHHHHT rot: HHHHHHTH flip: 0 2 4 6 => HHHTHTHT\n", + " 43: HHHTHTHT rot: HTHTHHHT flip: 01234567 => HTHTHTTT\n", + " 44: HTHTHTTT rot: TTHTHTHT flip: 01 45 => HHHTTHHT\n", + " 45: HHHTTHHT rot: HTTHHTHH flip: 01234567 => HHTTHTTT\n", + " 46: HHTTHTTT rot: TTHHTTHT flip: 0 2 4 6 => HHTTTHTT\n", + " 47: HHTTTHTT rot: HTTTHTTH flip: 01234567 => HHHTHHTT\n", + " 48: HHHTHHTT rot: THHHTHHT flip: 0123 => HHTHTTTT\n", + " 49: HHTHTTTT rot: TTTHHTHT flip: 01234567 => HHHHTTHT\n", + " 50: HHHHTTHT rot: TTHTHHHH flip: 0 2 4 6 => HHTTTTHT\n", + " 51: HHTTTTHT rot: HTHHTTTT flip: 01234567 => HHHHTHTT\n", + " 52: HHHHTHTT rot: HHHTHTTH flip: 01 45 => HTHTTHTT\n", + " 53: HTHTTHTT rot: THTTHTHT flip: 01234567 => HHTHHTHT\n", + " 54: HHTHHTHT rot: THHTHTHH flip: 0 2 4 6 => HHHTTTTT\n", + " 55: HHHTTTTT rot: TTTTTHHH flip: 01234567 => HHHHHTTT\n", + " 56: HHHHHTTT rot: HTTTHHHH flip: 012 456 => HHTTTTHT\n", + " 57: HHTTTTHT rot: TTTHTHHT flip: 01234567 => HHHHTHTT\n", + " 58: HHHHTHTT rot: HHTHTTHH flip: 0 2 4 6 => HHHHTTHT\n", + " 59: HHHHTTHT rot: THHHHTTH flip: 01234567 => HHTHTTTT\n", + " 60: HHTHTTTT rot: HTTTTHHT flip: 01 45 => HTHTTHTT\n", + " 61: HTHTTHTT rot: TTHTHTTH flip: 01234567 => HHTHHTHT\n", + " 62: HHTHHTHT rot: HHTHHTHT flip: 0 2 4 6 => HHHTTTTT\n", + " 63: HHHTTTTT rot: TTTTHHHT flip: 01234567 => HHHHHTTT\n", + " 64: HHHHHTTT rot: THHHHHTT flip: 012345 => HTTTTTTT\n", + " 65: HTTTTTTT rot: TTTTHTTT flip: 01234567 => HHHHHHHT\n", + " 66: HHHHHHHT rot: HHHHHHTH flip: 0 2 4 6 => HHHTHTHT\n", + " 67: HHHTHTHT rot: HTHTHHHT flip: 01234567 => HTHTHTTT\n", + " 68: HTHTHTTT rot: THTHTTTH flip: 01 45 => HHHTHHTT\n", + " 69: HHHTHHTT rot: TTHHHTHH flip: 01234567 => HHTTTHTT\n", + " 70: HHTTTHTT rot: THTTHHTT flip: 0 2 4 6 => HHHTTHHT\n", + " 71: HHHTTHHT rot: HHHTTHHT flip: 01234567 => HHTTHTTT\n", + " 72: HHTTHTTT rot: THHTTHTT flip: 012 456 => HTHTHTTT\n", + " 73: HTHTHTTT rot: THTHTHTT flip: 01234567 => HHHTHTHT\n", + " 74: HHHTHTHT rot: HHHTHTHT 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", + " 76: HHHHHHHT rot: HTHHHHHH flip: 01 45 => HHHTTHHT\n", + " 77: HHHTTHHT rot: HTHHHTTH flip: 01234567 => HHTTHTTT\n", + " 78: HHTTHTTT rot: TTHHTTHT flip: 0 2 4 6 => HHTTTHTT\n", + " 79: HHTTTHTT rot: THHTTTHT flip: 01234567 => HHHTHHTT\n", + " 80: HHHTHHTT rot: HHTTHHHT flip: 0123 => HHHHHTTT\n", + " 81: HHHHHTTT rot: HHHHHTTT flip: 01234567 => HHHTTTTT\n", + " 82: HHHTTTTT rot: TTTTTHHH flip: 0 2 4 6 => HHTHHTHT\n", + " 83: HHTHHTHT rot: HHTHHTHT flip: 01234567 => HTHTTHTT\n", + " 84: HTHTTHTT rot: HTHTTHTT flip: 01 45 => HHTHTTTT\n", + " 85: HHTHTTTT rot: TTTHHTHT flip: 01234567 => HHHHTTHT\n", + " 86: HHHHTTHT rot: HHHHTTHT flip: 0 2 4 6 => HHTTTTHT\n", + " 87: HHTTTTHT rot: TTHTHHTT flip: 01234567 => HHHHTHTT\n", + " 88: HHHHTHTT rot: HTHTTHHH flip: 012 456 => HTHTTHTT\n", + " 89: HTHTTHTT rot: HTHTTHTT flip: 01234567 => HHTHHTHT\n", + " 90: HHTHHTHT rot: THHTHHTH flip: 0 2 4 6 => HHHHHTTT\n", + " 91: HHHHHTTT rot: HHHTTTHH flip: 01234567 => HHHTTTTT\n", + " 92: HHHTTTTT rot: TTTTTHHH flip: 01 45 => HHHHTTHT\n", + " 93: HHHHTTHT rot: HHTTHTHH flip: 01234567 => HHTHTTTT\n", + " 94: HHTHTTTT rot: THTTTTHH flip: 0 2 4 6 => HHHHTHTT\n", + " 95: HHHHTHTT rot: HHHTHTTH flip: 01234567 => HHTTTTHT\n", + " 96: HHTTTTHT rot: TTTHTHHT flip: 01234 6 => HHHTHHTT\n", + " 97: HHHTHHTT rot: HTTHHHTH flip: 01234567 => HHTTTHTT\n", + " 98: HHTTTHTT rot: TTTHTTHH flip: 0 2 4 6 => HHHTTHHT\n", + " 99: HHHTTHHT rot: HHTHHHTT flip: 01234567 => HHTTHTTT\n", + " 100: HHTTHTTT rot: TTTHHTTH flip: 01 45 => HHHTHTHT\n", + " 101: HHHTHTHT rot: THHHTHTH flip: 01234567 => HTHTHTTT\n", + " 102: HTHTHTTT rot: TTHTHTHT flip: 0 2 4 6 => HTTTTTTT\n", + " 103: HTTTTTTT rot: TTHTTTTT flip: 01234567 => HHHHHHHT\n", " 104: HHHHHHHT rot: HHHHHHHT flip: 012 456 => HTTTTTTT\n", - " 105: HTTTTTTT rot: TTTHTTTT flip: 01234567 => HHHHHHHT\n", + " 105: HTTTTTTT rot: TTHTTTTT 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" + " 107: HTHTHTTT rot: HTTTHTHT flip: 01234567 => HHHTHTHT\n", + " 108: HHHTHTHT rot: HTHHHTHT flip: 01 45 => HHHTHHTT\n", + " 109: HHHTHHTT rot: HHTHHTTH flip: 01234567 => HHTTTHTT\n", + " 110: HHTTTHTT rot: TTTHTTHH flip: 0 2 4 6 => HHHTTHHT\n", + " 111: HHHTTHHT rot: THHHTTHH flip: 01234567 => HHTTHTTT\n", + " 112: HHTTHTTT rot: HTTHTTTH flip: 0123 => HHTTTTHT\n", + " 113: HHTTTTHT rot: HTTTTHTH flip: 01234567 => HHHHTHTT\n", + " 114: HHHHTHTT rot: THHHHTHT flip: 0 2 4 6 => HHTHTTTT\n", + " 115: HHTHTTTT rot: TTHHTHTT flip: 01234567 => HHHHTTHT\n", + " 116: HHHHTTHT rot: HHHTTHTH flip: 01 45 => HTHTTHTT\n", + " 117: HTHTTHTT rot: HTTHTHTT flip: 01234567 => HHTHHTHT\n", + " 118: HHTHHTHT rot: HTHHTHHT flip: 0 2 4 6 => HHHTTTTT\n", + " 119: HHHTTTTT rot: HHHTTTTT flip: 01234567 => HHHHHTTT\n", + " 120: HHHHHTTT rot: TTHHHHHT flip: 012 456 => HHTHTTTT\n", + " 121: HHTHTTTT rot: TTTTHHTH flip: 01234567 => HHHHTTHT\n", + " 122: HHHHTTHT rot: HHHHTTHT flip: 0 2 4 6 => HHTTTTHT\n", + " 123: HHTTTTHT rot: TTTTHTHH flip: 01234567 => HHHHTHTT\n", + " 124: HHHHTHTT rot: TTHHHHTH flip: 01 45 => HHHHHTTT\n", + " 125: HHHHHTTT rot: HHHHTTTH flip: 01234567 => HHHTTTTT\n", + " 126: HHHTTTTT rot: TTTTTHHH flip: 0 2 4 6 => HHTHHTHT\n", + " 127: HHTHHTHT rot: THHTHHTH flip: 01234567 => HTHTTHTT\n", + " 128: HTHTTHTT rot: HTHTTHTT flip: 0123456 => HHTHTTHT\n", + " 129: HHTHTTHT rot: THTHHTHT flip: 01234567 => HHTHTTHT\n", + " 130: HHTHTTHT rot: TTHTHHTH flip: 0 2 4 6 => HHHHTTTT\n", + " 131: HHHHTTTT rot: HHHHTTTT flip: 01234567 => HHHHTTTT\n", + " 132: HHHHTTTT rot: THHHHTTT flip: 01 45 => HHTHTTHT\n", + " 133: HHTHTTHT rot: TTHTHHTH flip: 01234567 => HHTHTTHT\n", + " 134: HHTHTTHT rot: THTTHTHH flip: 0 2 4 6 => HHHHTTTT\n", + " 135: HHHHTTTT rot: TTTTHHHH flip: 01234567 => HHHHTTTT\n", + " 136: HHHHTTTT rot: THHHHTTT flip: 012 456 => HHTHTTHT\n", + " 137: HHTHTTHT rot: TTHTHHTH flip: 01234567 => HHTHTTHT\n", + " 138: HHTHTTHT rot: TTHTHHTH flip: 0 2 4 6 => HHHHTTTT\n", + " 139: HHHHTTTT rot: HHTTTTHH flip: 01234567 => HHHHTTTT\n", + " 140: HHHHTTTT rot: TTHHHHTT flip: 01 45 => HHHHTTTT\n", + " 141: HHHHTTTT rot: THHHHTTT flip: 01234567 => HHHHTTTT\n", + " 142: HHHHTTTT rot: HHHHTTTT flip: 0 2 4 6 => HHTHTTHT\n", + " 143: HHTHTTHT rot: THTTHTHH flip: 01234567 => HHTHTTHT\n", + " 144: HHTHTTHT rot: THTTHTHH flip: 0123 => HHHTHHHT\n", + " 145: HHHTHHHT rot: THHHTHHH flip: 01234567 => HTTTHTTT\n", + " 146: HTTTHTTT rot: THTTTHTT flip: 0 2 4 6 => HHHTHHHT\n", + " 147: HHHTHHHT rot: THHHTHHH flip: 01234567 => HTTTHTTT\n", + " 148: HTTTHTTT rot: TTHTTTHT flip: 01 45 => HHHTHHHT\n", + " 149: HHHTHHHT rot: HHTHHHTH flip: 01234567 => HTTTHTTT\n", + " 150: HTTTHTTT rot: THTTTHTT flip: 0 2 4 6 => HHHTHHHT\n", + " 151: HHHTHHHT rot: HHTHHHTH flip: 01234567 => HTTTHTTT\n", + " 152: HTTTHTTT rot: TTTHTTTH flip: 012 456 => HHHHHHHH\n" ] }, { @@ -1227,7 +1122,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.5.3" + "version": "3.7.2" } }, "nbformat": 4,