The missing dollar riddle is a famous riddle that involves an informal fallacy. It dates to at least the 1930s, although similar puzzles are much older. [1]
The missing dollar riddle is a famous riddle that involves an informal fallacy. It dates to at least the 1930s, although similar puzzles are much older. [1]
Although the wording and specifics can vary, the puzzle runs along these lines:
Three guests check into a hotel room. The manager says the bill is $30, so each guest pays $10. Later the manager realizes the bill should only have been $25. To rectify this, he gives the bellhop $5 as five one-dollar bills to return to the guests.
On the way to the guests' room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. As the guests are not aware of the total of the revised bill, the bellhop decides to just give each guest $1 back and keep $2 as a tip for himself, and proceeds to do so.
As each guest got $1 back, each guest only paid $9, bringing the total paid to $27. The bellhop kept $2, which when added to the $27, comes to $29. So if the guests originally handed over $30, what happened to the remaining $1?
There seems to be a discrepancy, as there cannot be two answers ($29 and $30) to the math problem. On the one hand it is true that the $25 in the register, the $3 returned to the guests, and the $2 kept by the bellhop add up to $30, but on the other hand, the $27 paid by the guests and the $2 kept by the bellhop add up to only $29.
The misdirection in this riddle is in the second half of the description, where unrelated amounts are added together and the person to whom the riddle is posed assumes those amounts should add up to 30, and is then surprised when they do not — there is, in fact, no reason why the (10 − 1) × 3 + 2 = 29 sum should add up to 30.
The exact sum mentioned in the riddle is computed as:
SUM = $9 (payment by Guest 1) +
$9 (payment by Guest 2) +
$9 (payment by Guest 3) +
$2 (money in bellhop's pocket)
The trick here is to realize that this is not a sum of the money that the three people paid originally, as that would need to include the money the clerk has ($25). This is instead a sum of a smaller amount the people could have paid ($9 × 3 people = $27), added with the additional money that the clerk would not have needed had they paid that smaller amount ($27 paid - $25 actual cost = $2). Another way to say this is, the $27 already includes the bellhop's tip. To add the $2 to the $27 would be to double-count it. So, the three guests' cost of the room, including the bellhop's tip, is $27. Each of the 3 guests has $1 in his pocket, totaling $3. When added to the $27 revised cost of the room (including tip to the bellhop), the total is $30.
To obtain a sum that totals to the original $30, every dollar must be accounted for, regardless of its location.
Thus, the sensible sum can be expressed in this manner:
$30 = $1 (inside Guest pocket) +
$1 (inside Guest pocket) +
$1 (inside Guest pocket) +
$2 (inside bellhop's pocket) +
$25 (hotel cash register)
This sum does indeed come out to $30.
To further illustrate why the riddle's sum does not relate to the actual sum, the riddle can be altered so that the discount on the room is extremely large. Consider the riddle in this form:
Three people check into a hotel room. The clerk says the bill is $30, so each guest pays $10. Later the clerk realizes the bill should only be $10. To rectify this, he gives the bellhop $20 to return to the guests. On the way to the room, the bellhop realizes that he cannot divide the money equally. As the guests didn't know the total of the revised bill, the bellhop decides to just give each guest $6 and keep $2 as a tip for himself. Each guest got $6 back: so now each guest only paid $4; bringing the total paid to $12. The bellhop has $2. And $12 + $2 = $14 so, if the guests originally handed over $30, what happened to the remaining $16?
Now it is more obvious that the question is quite unreasonable. One cannot simply add a couple of payments together and expect them to total an original amount of circulated cash.
More economically, money is accounted by summing together all paid amounts (liabilities) with all money in one's possession (assets). That abstract formula holds regardless of the relative perspectives of the actors in this exchange.
To illustrate the issue through equations:
1) 10 + 10 + 10 = 30
2) 10 + 10 + 10 = 25 + 2 + 3
3) 10 + 10 + 10 - 3 = 25 + 2 + 3 - 3 (adding -3 to both sides of the equation to cancel out the +3 on the right side)
4) 10 - 1 + 10 - 1 + 10 - 1 = 25 + 2
5) 9 + 9 + 9 = 25 + 2 (obs: tip to bellhop has already been paid)
6) 27 = 27
How the riddle is deceptive comes in line 7:
7) 9 + 9 + 9 = 25 + 2
8) 9 + 9 + 9 + 2 ≠ 25 (pushing +2 to the other side without inverting the sign)
9) 27 + 2 ≠ 25
10) 29 ≠ 25
How it should be:
7) 9 + 9 + 9 = 25 + 2
8) 9 + 9 + 9 -2 = 25 + 2 -2 (adding -2 to both sides of the equation to cancel the +2 on the right side, which means the bellhop returned the tip or gave a discount of $2)
9) 9 + 9 + 9 - 2 = 25
10) 27 - 2 = 25
11) 25 = 25
The puzzle should subtract the bellhop's tip from the $27 rather than add it.
Let n guests initially pay p dollars each. The manager refunds r, to which the bellhop gives back b to each guest.
Each guest ends up with a balance of b−p (a negative amount), the manager with np−r and the bellhop r−nb. Whereas the guests' total initial payment is np, the sum of their eventual expense and the bellhop's pilferage is n(p−b) + (r−nb) = np + r− 2nb.
The discrepancy noted is thus np− (np + r− 2nb) = 2nb−r. With the riddle's values, 2 × 3 × $1 − $5 = $1.
Other values such as r = $20 and b = $6 give an unremarkable discrepancy of 2 × 3 × $6 − $20 = $16. Alternatively, values where b = r/2n yield no discrepancy. [2]
There are many variants of the puzzle. Professor David Singmaster's Chronology of Recreational Mathematics [3] suggests these type of mathematical misdirection puzzles descended from a problem in an 18th-century arithmetic book, Francis Walkingame's Tutor's Assistant [4] which was published, and republished, from 1751 to 1860 where it appeared on page 185, prob. 116 in this form, "If 48 taken from 120 leaves 72, and 72 taken from 91 leaves 19, and 7 taken from thence leaves 12, what number is that, out of which, when you have taken 48, 72, 19, and 7, leaves 12?" Singmaster adds, "Though this is not the same as the withdrawal problems below, the mixing of amounts subtracted and remainders makes me think that this kind of problem may have been the basis of the later kind."
An 1880 misdirection is given as "Barthel sees two boxes at a jeweller's, priced at 100 and 200. He buys the cheaper one and takes it home, where he decides he really prefers the other. He returns to the jeweller and gives him the box back and says that the jeweller already has 100 from him, which together with the returned box, makes 200, which is the cost of the other box. The jeweller accepts this and gives Barthel the other box and Barthel goes on his way. Is this correct?"
A model more similar in style to the modern version was given by Cecil B. Read in his 1933 Mathematical Fallacies. His puzzle produces an extra dollar: A man puts $50 in the bank. Then on subsequent days he withdraws $20 leaving $30; then $15 leaving $15; then $9 leaving $6, and finally $6 leaving $0. But $30 + $15 + $6 = $51. Where did the extra dollar come from?
The actual solution to this riddle is to add correctly (correct time, correct person and correct location) from the bank point of view which in this case seems to be the problem:
From the owner point of view the correct solution is this:
The solution appears very obvious if the owner withdraws every day only $10 from $50. To add up 40 + 30 + 20 + 10 using the same pattern from above would be too obviously wrong (result would be $100).
The answer to the question, "Where did the extra dollar come from?” can be found from consecutively adding the bank rest from three different days. This way is correct only if the money owner withdraws every day exact half of the money. Then it will add up. ($25 + $12.50 + $6.25) + $6.25 = $50
Another entry from 1933, R. M. Abraham's Diversions and Pastimes (still available in a Dover version) poses a slightly different approach with this problem from page 16 (problem 61). "A traveller returning to New York found that he had only a ten-dollar postal money order, and that his train fare was seven dollars. The ticket clerk refused to accept the money order, so the traveller went across the road to a pawn shop and pawned it for seven dollars. On his way back to the station he met a friend, who, to save the traveller the trouble of returning to redeem the money order, bought the pawn ticket from him for seven dollars. The traveller then bought his ticket and still had seven dollars when he got to New York. Who made the loss?" David Darling in his The Universal book of Mathematics, [5] credits this as an earlier version of the three men in a hotel version above.
Even more similar is the English, The Black-Out Book by Evelyn August in 1939; What happened to the shilling?, pp. 82 & 213. Three girls each pay five shillings to share a room. The landlord refunds 5 shillings via the bellboy, who gives them each one and keeps two.
And one more from the same theme appears in an Abbott and Costello routine in which Abbott asks Costello for a fifty-dollar loan. Costello holds out forty dollars and says, "That's all I have." Abbott responds, "Fine, you can owe me the other ten."
The riddle is used by psychotherapist (Chris Langham) with his mathematician client (Paul Whitehouse) in episode 5 of the 2005 BBC comedy series Help . [6]
A variation, also involving shillings and three men in a restaurant who are overcharged, appears in the third volume of Jennifer Worth's Call the Midwife books, Farewell to the East End (2009). There, repairman Fred poses it to the midwives of Nonnatus House.
Another variation, replacing the guests with shepherds, the clerk with a troll, the dollars with sheep and the bellboy with the troll's son, appears in Dr. No by Percival Everett.
The Eastern Caribbean dollar is the currency of all seven full members and one associate member of the Organisation of Eastern Caribbean States (OECS). The successor to the British West Indies dollar, it has existed since 1965, and it is normally abbreviated with the dollar sign $ or, alternatively, EC$ to distinguish it from other dollar-denominated currencies. The EC$ is subdivided into 100 cents. It has been pegged to the United States dollar since 7 July 1976, at the exchange rate of US$1 = EC$2.70.
Roulette is a casino game which was likely developed from the Italian game Biribi. In the game, a player may choose to place a bet on a single number, various groupings of numbers, the color red or black, whether the number is odd or even, or if the number is high or low.
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of borrowed funds is less than the total amount of money paid to the lender.
A gratuity is a sum of money customarily given by a customer to certain service sector workers such as hospitality for the service they have performed, in addition to the basic price of the service.
The 15 puzzle is a sliding puzzle. It has 15 square tiles numbered 1 to 15 in a frame that is 4 tile positions high and 4 tile positions wide, with one unoccupied position. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically, respectively. The goal of the puzzle is to place the tiles in numerical order.
For a Few Dollars More is a 1965 Spaghetti Western film directed by Sergio Leone. It stars Clint Eastwood and Lee Van Cleef as bounty hunters and Gian Maria Volonté as the primary villain. German actor Klaus Kinski plays a supporting role as a secondary villain. The film was an international co-production between Italy, West Germany, and Spain. The film was released in the United States in 1967, and is the second instalment of what is commonly known as the Dollars Trilogy.
Killer sudoku is a puzzle that combines elements of sudoku and kakuro. Despite the name, the simpler killer sudokus can be easier to solve than regular sudokus, depending on the solver's skill at mental arithmetic; the hardest ones, however, can take hours to solve.
Dollar cost averaging (DCA) is an investment strategy that aims to apply value investing principles to regular investment. The term was first coined by Benjamin Graham in his 1949 book The Intelligent Investor. Graham writes that dollar cost averaging "means simply that the practitioner invests in common stocks the same number of dollars each month or each quarter. In this way he buys more shares when the market is low than when it is high, and he is likely to end up with a satisfactory overall price for all his holdings."
The Refunding Certificate was a type of interest-bearing banknote that the United States Treasury issued in 1879. They issued it only in the $10 denomination, depicting Benjamin Franklin. Their issuance reflects the end of a coin-hoarding period that began during the American Civil War, and represented a return to public confidence in paper money.
Fractional currency, also referred to as shinplasters, was introduced by the United States federal government following the outbreak of the Civil War. These low-denomination banknotes of the United States dollar were in use between August 21, 1862, and February 15, 1876, and issued in denominations of 3, 5, 10, 15, 25, and 50 cents across five issuing periods. The complete type set below is part of the National Numismatic Collection, housed at the National Museum of American History, part of the Smithsonian Institution.
Merv Griffin's Crosswords is an American game show based on crossword puzzles. The show was created by its namesake, Merv Griffin, who died shortly after beginning production on the series. Ty Treadway was the host, and Edd Hall was the announcer.
The Cocoanuts is a musical with music and lyrics by Irving Berlin and a book by George S. Kaufman, with additional text by Morrie Ryskind.
Blue Toad Murder Files is a murder mystery puzzle video game developed and published by the British video game developer Relentless Software. The game is episodic and the first installment was released for the PlayStation 3 via the PlayStation Store in December 2009. The entire "season" of the game, Blue Toad Murder Files: The Mysteries of Little Riddle, includes six separate episodes. The Windows version was available to download from online games distribution sites as well as from the official website in November 2010. Support for PlayStation Move was added in December 2010. An iOS version was released in April 2013.
Booking.com is one of the largest online travel agencies. It is headquartered in Amsterdam, and is a subsidiary of Booking Holdings. Booking.com provides lodging reservation services for approximately 3.4 million properties, including 475,000 hotels, motels, and resorts and 2.9 million homes, apartments in over 220 countries and territories and in over 40 languages. It also sells flights in 55 markets and tours and activities in more than 1,300 cities. Hotel partners have reported not being paid for months at a time, leaving small hoteliers out of pocket.
Wheel of Fortune is an American television game show created by Merv Griffin. The show has aired continuously since January 6, 1975. Contestants solve word puzzles, similar to those in hangman, to win cash and prizes determined by spinning a giant carnival wheel. The current version of the series, which airs in nightly syndication, premiered on September 19, 1983.
Monster Strike is a mobile Japanese role-playing physics game with elements of puzzle, strategy and cooperative multiplayer. It is developed by Mixi for iOS and Android platforms. The game was created by former Capcom game designer Yoshiki Okamoto. In Japan, its name is often shortened to MonSt. The game can be summed up as a cross between Nintendo's Pokémon and Capcom's Metal Walker; the former in that it centers around monster collecting and the latter in that battles take place using catapult mechanics.
Major League Baseball (MLB) has a luxury tax called the "Competitive Balance Tax" (CBT). In place of a salary cap, the competitive balance tax regulates the total sum of money a given team can spend on their roster. Salary caps are common across professional sports leagues in the United States. Without these measures, teams would not be restricted on the amount of money spent on players' salaries. Therefore, teams with greater funding or revenue would possess a competitive advantage in their ability to attract top talent via higher salaries.
Hit Me is a 1996 American crime film directed by Steven Shainberg starring Elias Koteas, Laure Marsac, and William H. Macy. The film is based on the novel A Swell-Looking Babe by Jim Thompson.
The 2022 Money in the Bank was the 13th annual Money in the Bank professional wrestling pay-per-view (PPV) and livestreaming event produced by WWE. It was held for wrestlers from the promotion's Raw and SmackDown brand divisions. The event took place on Saturday, July 2, 2022, at the MGM Grand Garden Arena in the Las Vegas suburb of Paradise, Nevada; it was originally scheduled to be held at the city's Allegiant Stadium. This was the first Money in the Bank to be held on a Saturday, and it also elevated Money in the Bank to be regarded as one of WWE's five biggest events of the year, alongside the traditional "big four" of the Royal Rumble, WrestleMania, SummerSlam, and Survivor Series. This was also the final WWE pay-per-view and livestreaming event to take place before WWE owner Vince McMahon, who had served as chairman and chief executive officer of the company since 1982, announced his retirement on July 22, although he did return as Executive Chairman on January 10, 2023.