#exact change

Challenge #04671-L287: Fair Exchange

Unusual icon statues begin popping up all over the world. One puts in gold coins, and from the slot below, the exact amount of silver and copper coins comes out. The few times someone powerful enough to smash one managed to do so, they would find it not filled with coins at all. Merely more runes and sigils as it put itself back together again. -- Anon Guest

They were odd little altars. A figure of a Ravenfolk with an open beak clutching a bowl with its foot-talons. The wing-hands each held the image of a silver and copper coin. There was no explanation to their purpose, but they appeared at common crossroads in the city.

All over the city.

Of course the children tried it first. The beak was open like the image of the Ravenfolk expected to be fed something, and the toddlers and kindergartners of Luegovur were eager to supply. Stones and breadcrumbs passed into the mouth and out into the bowl. Bottlecaps went in, but copper coins came out. When copper coins went in, they passed through without changing.

What is the fewest number of coins you would need to make any given value of exact change from 1 to 99 cents?

one penny (1)

two pennies (2)

three pennies (3)

four pennies (4)

one nickel (1)

one nickel, one penny (2)

one nickel, two pennies (3)

one nickel, three pennies (4)

one nickel, four pennies (5)

one dime (1)

one dime, one penny (2)

one dime, two pennies (3)

one dime, three pennies (4)

one dime, four pennies (5)

one dime, one nickel (2)

one dime, one nickel, one penny (3)

one dime, one nickel, two pennies (4)

one dime, one nickel, three pennies (5)

one dime, one nickel, four pennies (6)

two dimes (2)

two dimes, one penny (3)

two dimes, two pennies (4)

two dimes, three pennies (5)

two dimes, four pennies (6)

one quarter (1)

one quarter, one penny (2)

one quarter , two pennies (3)

one quarter, three pennies (4)

one quarter, four pennies (5)

one quarter, one nickel (2)

one quarter, one nickel. one penny (3)

one quarter, one nickel, two pennies (4)

one quarter, one nickel, three pennies (5)

one quarter, one nickel, four pennies (6)

one quarter, one dime (2)

one quarter, one dime, one penny (3)

one quarter, one dime, two pennies (4)

one quarter, one dime, three pennies (5)

one quarter, one dime, four pennies (6)

one quarter, one dime, one nickel (3)

one quarter, one dime, one nickel, one penny (4)

one quarter, one dime, one nickel. two pennies (5)

one quarter, one dime, one nickel, three pennies (6)

one quarter, one dime, one nickel, four pennies (7)

one quarter, two dimes (3)

one quarter, two dimes, one penny (4)

one quarter, two dimes, two pennies (5)

one quarter, two dime, three pennies (6)

one quarter, two dimes, four pennies (7)

two quarters (2)

two quarters, one penny (3)

two quarters, two pennies (4)

two quarters, three pennies (5)

two quarters, four pennies (6)

two quarters, one nickel (3)

two quarters, one nickel, one penny (4)

two quarters, one nickel, two pennies (5)

two quarters, one nickel, three pennies (6)

two quarters, one nickel, four pennies (7)

two quarters, one dime (3)

two quarters, one dime, one penny (4)

two quarters, one dime, two pennies (5)

two quarters, one dime, three pennies (6)

two quarters, one dime, four pennies (7)

two quarters, one dime, one nickel (4)

two quarters, one dime, one nickel, one penny (5)

two quarters, one dime, one nickel, two pennies (6)

two quarters, one dime, one nickel, three pennies (7)

two quarters, one dime, one nickel, four pennies (8)

two quarters, two dimes (4)

two quarters, two dimes, one penny (5)

two quarters, two dimes, two pennies (6)

two quarters, two dimes, three pennies (7)

two quarters, two dimes, four pennies (8)

three quarters (3)

three quarters, one penny (4)

three quarters, two pennies (5)

three quarters, three pennies (6)

three quarters, four pennies (7)

three quarters, one nickel (4)

three quarters, one nickel, one penny (5)

three quarters, one nickel, two pennies (6)

three quarters, one nickel, three pennies (7)

three quarters, one nickel, four pennies (8)

three quarters, one dime (4)

three quarters, one dime, one penny (5)

three quarters, one dime, two pennies (6)

three quarters, one dime, three pennies (7)

three quarters, one dime, four pennies (8)

three quarters, one dime, one nickel (5)

three quarters, one dime, one nickel, one penny (6)

three quarters, one dime, one nickel, two pennies (7)

three quarters, one dime, one nickel, three pennies (8)

three quarters, one dime, one nickel, four pennies (9)

three quarters, two dimes (5)

three quarters, two dimes, one penny (6)

three quarters, two dimes, two pennies (7)

three quarters, two dimes, three pennies (8)

three quarters, two dimes, four pennies (9)

The maximum number of coins you would need to be able to make exact change at all times is three quarters, two dimes, one nickel, and four pennies (10)

To make all 99 perfect permutations, 80 include pennies, 75 include quarters, 60 include dimes, and 40 include a nickel.  4 give you exactly one coin, 9 give you two coins, 14 give you three coins, 18 give you four, 19 give you five, 16 give you six, 11 give you seven, 6 give you eight, and 2 give you nine.  You are statistically most likely to end up with between four and six coins in your change (53.535% chance of four, five or six coins)

In total, to make all values of exact change, you would need 470 coins; 200 pennies, 40 nickels, 80 dimes, and 150 quarters.

200 pennies weigh 500 grams.

40 nickels weigh 200 grams

80 dimes weigh 181.44 grams

150 quarters weigh 850.5 grams

In total, all these coins would weigh 1731.94 grams (1.73194 kilograms, 3.818274104 pounds), worth a combined 4950 cents, $49.50.  This comes out to a whopping 2.858066677 cents per gram; $28.58 per kilogram, $12.96 per pound.

There’s a 20/47 chance that any given value of exact change will contain at least one penny (42.55319149%)

4/47 that is contains a nickel (8.510638298%)

8/47 it contains a dime (17.0212766%)

15/47 it contains a quarter (31.91489362%)

If we compensate for the likelihood of receiving any given coin, we can refine this value even further.

200 pennies x (20/47) = 85.10638298 pennies = 85.10638298 cents

40 nickels x (4/47) = 3.404255319 nickels = 17.0212766 cents

80 dimes x (8/47) = 13.61702128 dimes = 136.1702128 cents

150 quarters x (15/47) = 47.87234043 quarters = 1196.808511 cents

Total: 150 coins, worth 1435.106383 cents ($14.35)

Even though pennies are more common, quarters make up the vast majority of the value of loose change because they’re worth 25 times as much while only weighing twice as much.

~85.1 pennies weigh 212.7659574 grams (0.4 cents per gram)

~3.4 nickels weigh 17.0212766 grams (1 cent per gram)

~13.6 dimes weigh 30.88340426 grams (4.409171076 cents per gram)

~47.9 quarters weigh 271.4361702 grams (4.409171076 cents per gram)

Total: 150 total coins weigh 532.1068085 grams (2.697026913 cents per gram)

So, after accounting for the value, mass, and probability of each coin, we can definitively conclude that a random pile of loose change will be worth an average: $26.97 cents per kilogram ($12.23 per pound)

We just want exact change.  Ripping a hole in our pockets is a service provided by your helpful president.  Balanced cash drawers are the impossibility imposed by Donald Trump’s petty penny removal.  Pinching the coin associated with that term is the opposite of frugal.

You can’t get exactly how much money you’re owed here in Soviet Russia.  The precise return of rubles is a matter of principle, which is why Trump doesn’t understand the problem.  It’d be one thing if practical results followed.  But the irksome reality matches mucky concepts.

Rounding to the nearest cent isn’t even an option anymore.  Sellers must go to the nearest nickel in the reboot to acknowledge inflation that’s totally at bay.  The cost to businesses skipping to the nearest amount will be exponentially more.  Cherish the halcyon days of Office Space joking about rounding to one-hundredth that in turn honored the one memorable notion presented in Superman III.  This new franchise has tired ideas.

Real savings take the form of battling entitlements.  The fearless leader and best businessman is too craven to confront the generational ripoff of Social Security, too egomaniacal to get government out of the retirement business, and too insecure to think workers should be allowed to invest their own money as opposed to letting politicians spend it before the workday is done.  Cutting millions when trillions are needed is such a Trump move that it could be covered in tacky black glass.

Waging war on numismatics is a loss even with an easy win.  The obsessive drive to eliminate the most trifling costs resembles waste, fraud, and abuse posturing.  Trump acts as if nobody had thought to make things more efficient before.  The alleged antidote to traditional politicians is as hackneyed as his contemptuous CEO character.

As a reminder, those millions helped.  Producing square paper and round metal that can be used to pay off debts is one of government’s actually justified expenditures.  The government has a quite limited number of legitimate functions which includes providing currency.  Spending money so we can have some is an expenditure even road-privatizing libertarians could concede is one of the few federal actions that’s justified.

Dividing a dollar into a hundred parts offends math’s enemies.  Spot them by how they refuse to admit which direction the debt clock spins.  By the way, does anyone have four quadrillion pennies?  Check the dish near the register.

Trying to figure out if they should round up or down is an additional gift from a president whose tax on purchases has already made commerce burdensome.  Tariffs have made prices so high that change is irrelevant.  The notion is possibly not a winning political or economic strategy over time.  We’re simply not allowed the easy solution involving having the precise amount.

Abolishing cents makes no sense.  Saving a few billion of the little guys completely misses the point in the classic style of this repeat presidency.  There’s no reason to endure agony if it doesn’t result in learning from it.

Figuring what the market wants isn’t the ostensible titan’s specialty.  The business genius who slapped his name on mail-order steaks, pedestrian bottled water, vodka you wouldn’t serve a hobo in hospice, a board game not one human has ever played, and a million other shoddy piles of dreck that he tried to con you into believing were worth premium pricing because he blessed the labels with his saintly family name is all about creating value.  That’s why he respects almost all the parts of a dollar.

Uninterested consumers are free to leave pennies on the counter or sidewalk.  Fanciful individuals who enjoy making wishes can collect them for fountain placement.  And those interested in precision can ensure they pay the exact amount.  Chaos without thinking ahead makes life exciting.  America has become an Ollie’s trip where we don’t know what baffling goods we’ll find.  The outcome is unplanned like everything else a self-proclaimed visionary imposes.

The smallest change makes a big deal.  We’re not just talking about what used to rattle in pockets.  Having precise payment is a small satisfaction that’s been revoked by a clumsy oaf who’s regrettably authorized to oversee the mint.

Physical media collectors want the option.  The simple act of churning out every denomination makes life easy for those buying with bills and coins.  The only thing it would’ve took to keep it going was doing nothing.  Naturally, Trump couldn’t control himself.

The bad and good thing about acting like a tyrant is the next person can take it back.  Trump delights in issuing commands like Caesar, who’s the namesake of an Atlantic City casino that’s still open.  But an executive interested in maintaining the republic’s boundaries can act similarly, only in reverse.  We can hope the next president reverses a shortsighted effort to make the smallest amount extinct.

To recognize the person who tried to dump all of the most modest way to complete a purchase, put Trump on the coin.  He’ll think he’s being honored.  Refusing to make them gold-colored won’t be enough of a clue.  That would involve removing Abe Lincoln from the most common way he appears in the public consciousness.  But Trump’s already tried to do that.