Update explanation of modular exponentiation

README.md

@@ -142,7 +142,7 @@ For each notebook you can hover on the title to see a description, or click the
 | [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/RiddlerLottery.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FRiddlerLottery.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FRiddlerLottery.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/RiddlerLottery.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/RiddlerLottery.ipynb) | 2019 | <a href="ipynb/RiddlerLottery.ipynb" title="Can you find what lottery number tickets these five friends picked?">Lottery</a> |
 | [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/NightKing.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FNightKing.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FNightKing.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/NightKing.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/NightKing.ipynb) | 2019 | <a href="ipynb/NightKing.ipynb" title="A battle between the army of the dead and the army of the living">How Many Soldiers to Beat the Night King?</a> |
 | [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/Mean%20Misanthrope%20Density.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FMean%20Misanthrope%20Density.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FMean%20Misanthrope%20Density.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/Mean%20Misanthrope%20Density.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/Mean%20Misanthrope%20Density.ipynb) | 2017 | <a href="ipynb/Mean%20Misanthrope%20Density.ipynb" title="How crowded will this neighborhood be, if nobody wants to live next door to anyone else?">Misanthropic Neighbors</a> |
-| [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/Orderable%20Cards.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FOrderable%20Cards.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FOrderable%20Cards.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/Orderable%20Cards.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/Orderable%20Cards.ipynb) | 2018 | <a href="ipynb/Orderable%20Cards.ipynb" title="Can you get your hand of cards into a nice order with just one move?">Properly Ordered Card Hands</a> |
+| [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/Orderable%20Cards.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FOrderable%20Cards.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FOrderable%20Cards.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/Orderable%20Cards.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/Orderable%20Cards.ipynb) | 2018 | <a href="ipynb/Orderable%20Cards.ipynb" title="Can you get your hand of cards into a nice order with just one move?">Properly Organized Card Hands</a> |
 | [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/RaceTrack.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FRaceTrack.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FRaceTrack.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/RaceTrack.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/RaceTrack.ipynb) | 2021 | <a href="ipynb/RaceTrack.ipynb" title="Race virtual cars around a circular track defined on a grid of points.">Race Track</a> |
 | [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/SplitStates.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FSplitStates.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FSplitStates.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/SplitStates.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/SplitStates.ipynb) | 2021 | <a href="ipynb/SplitStates.ipynb" title="Split the US states into two near-halves by area.">Split the States</a> |
 | [C](https://colab.research.google.com/github/norvig/pytudes/blob/main/ipynb/TourDe538.ipynb) [D](https://beta.deepnote.org/launch?template=python_3.6&url=https%3A%2F%2Fgithub.com%2Fnorvig%2Fpytudes%2Fblob%2Fmain%2Fipynb%2FTourDe538.ipynb) [M](https://mybinder.org/v2/gh/norvig/pytudes/main?filepath=ipynb%2FTourDe538.ipynb) [N](https://nbviewer.jupyter.org/github/norvig/pytudes/blob/main/ipynb/TourDe538.ipynb) [S](https://studiolab.sagemaker.aws/import/github/norvig/pytudes/blob/main/ipynb/TourDe538.ipynb) | 2020 | <a href="ipynb/TourDe538.ipynb" title="Solve a puzzle involving the best pace for a bicycle race.">Tour de 538</a> |

ipynb/Orderable Cards.ipynb

@@ -29,7 +29,7 @@
     "|*N*|Concrete Hands|Abstract Hands|\n",
     "|--|---|---|\n",
     "|6|20,358,520|4,096|\n",
-    "13|635,013,559,600|67,108,864|\n",
+    "|13|635,013,559,600|67,108,864|\n",
     "|*N*|(52 choose *N*) | *N*<sup>4</sup>|\n",
     "\n",
     "That's a big improvement! Now let's start designing a program to solve the problem:\n",

ipynb/TruncatablePrimes.ipynb

@@ -13,11 +13,11 @@
     "\n",
     "[![](https://community.wolfram.com//c/portal/getImageAttachment?filename=IMG_20181212_120939.jpg&userId=143131)](https://community.wolfram.com/groups/-/m/t/1569707)\n",
     "\n",
-    "The 24-digit number printed on it is a prime, and as you sharpen the pencil and remove digits one at a time from the left, the resulting numbers are all primes:\n",
+    "The 24-digit number printed on this pencil is a prime, and as you sharpen the pencil and remove digits one at a time from the left, the resulting numbers are all primes:\n",
     "\n",
     "    357686312646216567629137 is prime\n",
     "     57686312646216567629137 is prime\n",
-    "      7686312646216567629137 is prime|\n",
+    "      7686312646216567629137 is prime\n",
     "                 ...\n",
     "                         137 is prime\n",
     "                          37 is prime\n",
@@ -1127,7 +1127,7 @@
     "\n",
     "# Note on Modular Exponentiation\n",
     "\n",
-    "Just one more thing: none of this would work unless we can efficiently compute *a*<sup>(*n* - 1)</sup> (mod *n*).  How does the `pow` builtin function do it? When *a* and *n* are 24-digit numbers, if we naively tried to compute `a ** (n - 1)`, we'd have two problems: we'd need nearly a billion petabytes to store the result, and we'd need centuries to compute it. The way around these problems is to use [modular exponentiation](https://en.wikipedia.org/wiki/Modular_exponentiation) where we apply the modulus to each intermediate result, and cut the exponent in half each iteration, so we need only do *O*(log *n*) multiplications, not *O*(*n*). That's a big difference: 10<sup>24</sup> is a trillion trillion, and log<sub>2</sub>(10<sup>24</sup>) is only 80. \n",
+    "Just one more thing: none of this would work unless we can efficiently compute *a*<sup>(*n* - 1)</sup> (mod *n*).  How does the `pow` builtin function do it? When *a* and *n* are 24-digit numbers, if we naively tried to compute `a ** (n - 1)` and then apply (mod *n*), we'd have two problems: we'd need nearly a billion petabytes to store the result, and we'd need centuries to compute it. The way around these problems is to use [modular exponentiation](https://en.wikipedia.org/wiki/Modular_exponentiation) where we apply the modulus to each intermediate result, and cut the exponent in half each iteration, so we need only do *O*(log *n*) multiplications, not *O*(*n*). That's a big difference: 10<sup>24</sup> is a trillion trillion, and log<sub>2</sub>(10<sup>24</sup>) is only 80. \n",
     "\n",
     "There are two key ides that make this work:\n",
     "1) *b*<sup>2*e*</sup> = (*b* × *b*)<sup>*e*</sup>\n",

py/pytudes.py

@@ -98,7 +98,7 @@ notebooks = {
 ('Lottery', 2019, 'RiddlerLottery.ipynb',  'Can you find what lottery number tickets these five friends picked?'),
 ('How Many Soldiers to Beat the Night King?', 2019, 'NightKing.ipynb',  'A battle between the army of the dead and the army of the living'),
 ('Misanthropic Neighbors', 2017, 'Mean%20Misanthrope%20Density.ipynb', 'How crowded will this neighborhood be, if nobody wants to live next door to anyone else?'),
-('Properly Ordered Card Hands', 2018, 'Orderable%20Cards.ipynb', 'Can you get your hand of cards into a nice order with just one move?'),
+('Properly Organized Card Hands', 2018, 'Orderable%20Cards.ipynb', 'Can you get your hand of cards into a nice order with just one move?'),
 ('Race Track', 2021, 'RaceTrack.ipynb', 'Race virtual cars around a circular track defined on a grid of points.'),
 ('Split the States', 2021, 'SplitStates.ipynb', 'Split the US states into two near-halves by area.'),
 ('Tour de 538', 2020, 'TourDe538.ipynb', 'Solve a puzzle involving the best pace for a bicycle race.'),