diff --git a/ipynb/Diamonds.ipynb b/ipynb/Diamonds.ipynb index ea4ed07..cfeee21 100644 --- a/ipynb/Diamonds.ipynb +++ b/ipynb/Diamonds.ipynb @@ -7,7 +7,7 @@ "source": [ "
Peter Norvig
May 2023
\n", " \n", - "# Wei-Hwa's Diamond-Buying Game \n", + "# The Diamond Game: A Probability Puzzle\n", "\n", "Puzzle master [**Wei-Hwa Huang**](https://en.wikipedia.org/wiki/Wei-Hwa_Huang) poses this problem (reworded by me): \n", " \n", @@ -566,7 +566,7 @@ "id": "6b454a23-6c5b-4882-8c3c-b2c17c6caac2", "metadata": {}, "source": [ - "The plot below looks at situations where (on the x-axis) you have a certain number of coins before you buy a bag, and on the y-axis there turns out to be a certain number of diamonds in the bag you bought. A blue dot means that the optimal strategy is to sell the bag back in that situation. This is the **true** optimal strategy, obtained by having exact knowledge of the stockpile, something that a player could never obtain exactly." + "The plot below looks at situations where the x-axis lists the number of coins you have before you buy a bag, and the y-axis lists the number of diamonds in the bag you bought. A blue dot at position (x, y) means that the optimal strategy is to sell the bag back in that situation. This is the *true* optimal strategy for `stock10`, obtained by having exact knowledge of the stockpile, something that a player could never obtain exactly." ] }, { @@ -595,7 +595,10 @@ "id": "1256a5ec-dd6b-4f4a-8d35-50d5456c0a8f", "metadata": {}, "source": [ - "This plot shows that, for this stockpile, when you have a multiple of 12 coins, it is usually best to keep the bag you just bought, no matter how bad the bag. For example, with 36 coins it is always better to use the 36 coins to buy and keep 3 bags than to sell one back and be left with only enough coins to buy 2 bags. With each extra coin from 37 to 47, we can afford to be more picky, and the bag values that we will sell back increase, from 93 up to 107. There are a few cases where this rule fails to hold–with 120 coins, sell back a bag with 69 or fewer diamonds. (Such bags are more than 3 standard deviations from the mean, so should occur less than 0.5% of the time.)\n", + "We notice three things about this plot:\n", + "- It is made out of sawtooths.\n", + "- The edge of each sawtooth goes up as the number of coins mod 12 goes up. When you have an exact multiple of 12 coins, it is (almost always) best to keep the bag you just bought, no matter how bad the bag. For example, with 24 coins it is always better to buy and keep 2 bags than to sell one back and be left with only enough coins to keep 1 bag. With each extra coin from 25 to 35, we can afford to be more picky, and the bag values that we will sell back increase, from 96 up to 109. (The first time that we see a multiple of 12 where it is better to sell a bag back is with 120 coins, and then only for bags with 69 or fewer diamonds. Such bags are more than 3 standard deviations from the mean, so they occur less than 0.5% of the time.)\n", + "- The height of each sawtooth goes down as the number of coins increases. I think that is because with, say, 23 coins you know you have 11 chances to buy and sell back the next bag, so you can afford to reject pretty good bags, such as 100 diamonds. With 119 coins you also have 11 extra coins that can be used to buy and sell back, but you need to allocate them over nine different bags that you will be keeping, so you don't want to use them all up right away.\n", "\n", "# Evaluating a strategy with a metric\n", "\n", @@ -701,7 +704,7 @@ "source": [ "# Visualizing Expected Values\n", "\n", - "The function `plot_E` below allows us to plot the expected value for a strategy as a function of the number of coins. We can compare different stockpiles, strategies, and/or numbers of peeks. The function produces two plots: one that displays the expected total number of diamonds as a function of the number of coins, and one that displays the expected number of diamonds per 12 coins. " + "The function `plot_E` below allows us to plot the expected value for a strategy as a function of the number of coins. We can compare different stockpiles, strategies, and/or numbers of peeks. The function produces two plots: one that displays the expected total number of diamonds as a function of the number of coins, and one that displays the expected number of diamonds per 12 coins. (You can think of the second plot as being the same as the first, but rotated so that the x-axis parallels the values where the number of coins mod 12 is zero.)" ] }, {