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Peter Norvig
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116
Coin Flip.ipynb
116
Coin Flip.ipynb
@@ -337,7 +337,11 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"That's a 15-move sequence that is guaranteed to lead to a win. Do I believe it? It does appear to work. A colleague did the puzzle and got the same answer. And here's further validation: The function `random_devil` takes a sequence of moves and plays those moves with a devil that chooses randomly:"
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"That's a 15-move sequence that is guaranteed to lead to a win. \n",
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"\n",
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"# Verifying the Solution\n",
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"\n",
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"Do I believe the solution is correct? Playing with paper and pencil, it does appear to work. A colleague did the puzzle and got the same answer. And here's further validation: The function `random_devil` takes a sequence of moves and plays those moves with a devil that chooses randomly, returning the number of moves it takes until the player wins:"
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]
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},
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{
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@@ -376,22 +380,22 @@
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{
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"data": {
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"text/plain": [
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"Counter({0: 969,\n",
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" 1: 1034,\n",
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" 2: 966,\n",
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" 3: 1026,\n",
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" 4: 975,\n",
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" 5: 979,\n",
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" 6: 987,\n",
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" 7: 1047,\n",
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" 8: 978,\n",
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" 9: 979,\n",
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" 10: 998,\n",
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" 11: 1033,\n",
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" 12: 1025,\n",
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" 13: 973,\n",
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" 14: 1038,\n",
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" 15: 993})"
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"Counter({0: 1037,\n",
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" 1: 1006,\n",
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" 2: 1011,\n",
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" 3: 998,\n",
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" 4: 980,\n",
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" 5: 976,\n",
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" 6: 1057,\n",
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" 7: 1043,\n",
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" 8: 963,\n",
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" 9: 1021,\n",
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" 10: 966,\n",
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" 11: 1012,\n",
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" 12: 995,\n",
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" 13: 975,\n",
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" 14: 990,\n",
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" 15: 970})"
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]
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},
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"execution_count": 11,
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@@ -409,8 +413,82 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This says that the player won all 16,000 times. If the player ever lost, there would be an entry for `None` in the Counter.\n",
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"The remarkable thing, which I can't explain, is that there are very nearly 1,000 results for each of the counts from 0 to 15. Can you explain that?"
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"This says that the player won all 16,000 times. (If the player ever lost, there would have been an entry for `None` in the Counter.)\n",
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"The remarkable thing, which I can't explain, is that there are very nearly 1,000 results for each of the counts from 0 to 15. Can you explain that?\n",
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"\n",
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"# Visualizing the Solution\n",
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"\n",
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"How does the solution work? One answer is \"it takes care of all possibilities.\" But it would be nice to gain more insight. I'll print a table showing the belief state after each move. But I will do two things to make the belief state easier to read. First, I will *canonicalize* the belief state to be independent of rotations. By that I mean that if the belief stae contains `{'HHHT', 'HHTH', 'HTHH', 'THHH'}`, which are all rotations of each other, I will only print one of them. (Arbitrarily, I choose the one that comes first alphabetically, `'HHHT'`.) This gets us down from 16 coin sequences in the belief state to six.\n",
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"Second, I will print each coin sequence in its own column, so that they line up and it is easy to see when a particular coin sequence disappears from the belief state"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"def show(moves):\n",
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" \"For each move, print the move number, move, and belief state.\"\n",
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" belief = initial_belief\n",
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" show_line(0, 'start', belief)\n",
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" for (i, move) in enumerate(moves, 1):\n",
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" belief = update(belief, move)\n",
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" show_line(i, move, belief)\n",
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" \n",
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"def join(items, sep=''): return sep.join(map(str, items))\n",
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" \n",
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"def canonical(belief): return sorted(set(min(rotations(coins)) for coins in belief))\n",
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"\n",
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"def show_line(i, move, belief, order=canonical(initial_belief)):\n",
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" \"Print the move number, move, and belief state.\"\n",
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" ordered_belief = [(coins if coins in belief else ' ')\n",
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" for coins in order]\n",
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" print('{:2} {:5} {}'.format(i, join(move), join(ordered_belief, ' ')))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" 0 start HHHH HHHT HHTT HTHT HTTT TTTT\n",
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" 1 0123 HHHH HHHT HHTT HTHT HTTT \n",
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" 2 02 HHHH HHHT HHTT HTTT TTTT\n",
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" 3 0123 HHHH HHHT HHTT HTTT \n",
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" 4 01 HHHH HHHT HTHT HTTT TTTT\n",
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" 5 0123 HHHH HHHT HTHT HTTT \n",
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" 6 02 HHHH HHHT HTTT TTTT\n",
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" 7 0123 HHHH HHHT HTTT \n",
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" 8 0 HHHH HHTT HTHT TTTT\n",
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" 9 0123 HHHH HHTT HTHT \n",
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"10 02 HHHH HHTT TTTT\n",
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"11 0123 HHHH HHTT \n",
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"12 01 HHHH HTHT TTTT\n",
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"13 0123 HHHH HTHT \n",
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"14 02 HHHH TTTT\n",
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"15 0123 HHHH \n"
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]
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}
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],
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"source": [
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"show(search())"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We can see that every odd-numbered move flips all four coins to eliminate the possibility of `TTTT`.\n",
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"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 flips a single coin, thus eliminating the \"three heads\" and \"one heads\" sequences, and 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 rid of the last possibility."
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]
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}
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],
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