Valuate a Bag of Coins

A bag of new US coins is stolen from a bank. Without opening the bag, what can be said about the monetary value of its contents? One obvious and easily measured physical characteristic is the bag's weight. Assuming a one-pound bag of coins, combine the Wolfram Knowledgebase's know-how on currencies and built-in ability to solve linear equations to study the expected value of the plunder.

To begin, return a list of US coins in current circulation by means of an implicitly defined entity class.

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EntityClass["CurrencyDenomination", {EntityProperty[ "CurrencyDenomination", "IssuingCountry"] -> Entity["Country", "UnitedStates"], EntityProperty["CurrencyDenomination", "Format"] -> "coin"}]

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Expand the implicitly defined entity class by clicking the [+], find its members, and sort them by value.

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coinsUS = EntityList[ EntityClass[ "CurrencyDenomination", { EntityProperty[ "CurrencyDenomination", "IssuingCountry"] -> Entity[ "Country", "UnitedStates"], EntityProperty["CurrencyDenomination", "Format"] -> "coin"}]] // SortBy[#[EntityProperty["CurrencyDenomination", "Value"]] &]

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Make a collage of coin images.

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ImageCollage[ EntityValue[coinsUS, EntityProperty["CurrencyDenomination", "Image"]], Background -> White]

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Summarize coin properties in a table.

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TextGrid[{ImageResize[#2, 60], #1, Row[Riffle[#3, Style[" | ", Gray]]]} & @@@ EntityValue[ coinsUS, {EntityProperty["CurrencyDenomination", "Entity"], EntityProperty["CurrencyDenomination", "Image"], EntityProperty["CurrencyDenomination", "PeopleOnCurrency"]}], Alignment -> {Left, Center}, Dividers -> All] // TraditionalForm

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Get coin denominations (in cents) and masses (in grams) and convert masses to rational numbers.

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{values, masses} = Transpose[EntityValue[coinsUS, {"Value", "Weight"}]]

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coinsandweights = Transpose[{ QuantityMagnitude[UnitConvert[values, "USCents"]], Rationalize[QuantityMagnitude[N[UnitConvert[masses, "Grams"]]]] }]

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lcm = LCM @@ Denominator[Rationalize[coinsandweights][[All, 2]]];

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rationalcoinweights = lcm #2 & @@@ Rationalize[coinsandweights]

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Find all coin distributions that are compatible with a weight measurement of one pound with an error measurement of ± 0.1% (assuming the bag itself contributes negligibly).

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meanWeight = QuantityMagnitude[UnitConvert[Quantity[1, "Pounds"], "Grams"]];

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error = Normal[Quantity[0.1, "Percent"]];

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{minScaledWeight, maxScaledWeight} = {Floor[lcm meanWeight (1 - error/2)], Ceiling[lcm meanWeight (1 + error/2)]}

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Use FrobeniusSolve to determine all possible coin collections giving the required total weight.

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allSolutions = Monitor[ Table[fb = FrobeniusSolve[rationalcoinweights, scaledweight], {scaledweight, minScaledWeight, maxScaledWeight}], Row[{NumberForm[ 100. (scaledweight - minScaledWeight)/(maxScaledWeight - minScaledWeight), {4, 1}, "NumberPadding" -> {" ", "0"}], "%: ", Length[fb], " solutions"}] ];

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Flatten[allSolutions, 1] // Length

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Find the minimum, median, mean, and maximum values of the total monetary value of the coins in the bag (assuming all combinations are equally likely).

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coins = coinsandweights[[All, 1]];

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dollarValues = (coins.#/100.) & /@ Flatten[allSolutions, 1];

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TextGrid[With[{stats = {Min, Median, Mean, Max}}, Transpose[{stats, NumberForm[Quantity[#, "USDollars"], {\[Infinity], 2}] & /@ Through[stats[dollarValues]]}]], Alignment -> {{Left, Decimal}}, Dividers -> All, Background -> {Automatic, {{LightBlue, None}}}] // TraditionalForm

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Make a histogram of the distribution of total money value.

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Histogram[dollarValues, Automatic, "PDF", AxesLabel -> {Quantity[None, "USDollars"], "fraction"}]

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The weight distribution of all sacks is fairly uniform.

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weights = coinsandweights[[All, -1]]; lbg = QuantityMagnitude[UnitConvert[Quantity[1, "Pounds"], "Grams"]]; weightvalues = (weights.#)/lbg & /@ Flatten[allSolutions, 1];

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Histogram[weightvalues, 50, "PDF", AxesLabel -> {Quantity[None, "USDollars"], "fraction"}]

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Plot the distribution of the number of coins.

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Histogram[Total /@ Flatten[allSolutions, 1], {5}]

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The bivariate distribution of the monetary value versus the number of coins in the sack.

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Histogram3D[{coins.#/100., Total[#]} & /@ Flatten[allSolutions, 1], AxesLabel -> {Quantity[None, "USDollars"], "coins"}]

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