« return to puzzle

Let Me Tell You The Story

Kevin Wald

One must first determine how much money each of the five characters has when. These characters are, in the order given in boldface in the last paragraph:

(For convenience, we shall call these characters E, B, V, L, and C, below.)

Other costs: The telephone call at the end costs 50 cents. Since the end of the story is the tale of Charlie on the MTA, the cost for getting on at Kendall is 10 cents and the exit fee is 5 cents. Thoughts, of course, cost a penny.

The sequence of financial transactions is as follows:

  (1, 7:00 PM) E gives V a penny for her thoughts.
  (2, a bit later) V hires E, E gets two dances.
  (3, a bit later) E takes the T, but does not pay the exit fee.
  (4, 10:00 PM) C pays B for PJs.
  (5, midnight) B hires E.
  (6, a bit later) C pays L for a session.
  (7, a bit later) B pays L for a session.
  (8, 10:00 AM) L makes change (a "danceworth") for V.
  (9, a bit later) L hires E.
  (10, afternoon) C pays V for a dance.
  (11, at least two hours later) V pays B for PJs.
  (12, 5:00 PM) B gives E a penny for his thoughts.
  (13, a bit later) E pays for a phone call.

Taking the characters one by one:

V: At the end, V has 6 cents (the cost of 3 glasses of two cents plain, or 2 tickets to the Three-Penny Opera). Since at the start, nobody has "two of anything that matched," there are at most 5 pennies in the world, so V has a nickel and a penny.

In the course of the story, V gets 1 cent from E (1), hires him for 25 - 2*10 = 5 cents (2), makes 10 cents change with L (8), gets 10 cents from C (10), and pays B 2 * 7.5 = 15 cents (11). Thus, at the start, V has 6 + 15 - 10 + 5 - 1 = 15 cents. The only way to get this with nothing "matching" is a dime and a nickel.

Transactions determined: She gets a penny in (1), and pays a nickel in (2) and breaks a dime into two nickels in (8) (if she'd paid a dime in (2) and gotten change, she'd have nothing to "break in two" in (8)). We don't yet know the denominations for (10) and (11), but right before (10) she has two nickels and a penny.

E: At midnight, E is completely broke. Prior to this, he pays V a penny, (1), is hired by her for a nickel (1), and spends 10 cents on the T (3), so at the start E has 10 - 5 + 1 = 6 cents, which must be a nickel and a penny. After midnight, E gets 25 cents from B (5), then 25 cents from L (9), gets 1 cent from B (12), and spends 50 cents on a phone call (13), so at the end E has 25 + 25 + 1 - 50 = 1 cent; that is, a penny.

Transactions determined: In (3) he pays the two nickels he has. In (13) he pays two quarters, so he gets one quarter in each of (5) and (9). In (12) he gets a penny.

C: At the end, C has only 10 cents ("exactly what he needs to get on"). In the course of the story, C pays B 2 * 7.5 = 15 cents (4), pays L 5 cents (6), and pays V 10 cents (10), so at the start C had 10 + 10 + 5 + 15 = 40 cents, which (to avoid duplicates) must have been a quarter, a dime and a nickel.

Since E's fee from B in (5), a quarter, has a 50-50 chance of having come from C, in (4) C pays B a quarter (and possibly more coins) and gets change; since B at this point has no duplicate coins, the only possible combination is that C pays B exactly a quarter and gets a dime in change. So after (4) C has two dimes and a nickel. He pays L a nickel in (6) (he "handed her something," so there is no changemaking involved), after which C has two dimes. To finish off C, we examine B.

Transaction determined: C pays L a nickel in (6).

B: Right before B hires E in (5), she has two quarters (to make it a fifty-fifty chance that the quarter she gives E comes from C), the nickel she will give L in (7) (half of the change from (8)), and the penny she will give E in (12); she does not have a dime, or else she would have had two dimes before she gives C a dime change in (4). So at the start, she has one quarter, one dime, one nickel, and one penny (and nothing of higher denomination, or else she would have had more than a third of the whole pot rather than less).

After B hires E for a quarter in (5), and pays L a nickel in (7), B has a quarter and a penny. So before the (10) and (11):

So for (11), B can't make any change, so V must give her a dime and a nickel (no other exact 15-cent combination can be made out of what V and C have), and (10) consists of C giving V a dime. So C has a dime at the end.

After (11), B has a quarter, a dime, a nickel, and a penny; she gives E a penny in (12), leaving her with a quarter, a dime, and a nickel at the end.

Transactions determined: In (4) B gets a quarter from C and gives him a dime. In (7) B gives L a nickel. In (10) C gives V a dime. In (11) V gives B a dime and a nickel.

L: The quarter L gives E in (9) is "ninety-odd" percent of the money in her purse (which does not include the dime she got from V in (8)), so the other money in the purse is non-zero but no more than 25/9 = 2.78 cents. L can't have two pennies in her purse (or else she would have had them since the beginning), so it was one penny. In (8) got a dime from V and gave her two nickels; one from C in (6) and one from B in (7). Thus, at the start L had a quarter and a penny, and at the end L had a dime and a penny.

The complete history of everyone's finances (where the four-digit sequnce QDNP indicates the number of quarters, dimes, nickels, and pennies each character has after the given transaction):

(1) E gives V a penny for her thoughts. 00101111011110011110
(2) V hires E, E gets two dances. 00201111010110011110
(3) E takes the T. 00001111010110011110
(4) C pays B for PJs. 00002011010110010210
(5) B hires E. 10001011010110010210
(6) C pays L for a session. 10001011010110110200
(7) B pays L for a session. 10001001010110210200
(8) L makes change for V. 10001001002111010200
(9) L hires E. 20001001002101010200
(10) C pays V for a dance. 20001001012101010100
(11) V pays B for PJs. 20001111001101010100
(12) B gives E a penny for his thoughts. 20011110001101010100
(13) E pays for a phone call (right before [END]). 00011110001101010100

Note that at the start and at the end, each character has at most one quarter, dime, nickel, and penny (and no other money); thus, the quantities of these coins forms in each case a 4-bit binary number. (The final line of the story hints at the 4-bit coding.) These numbers, translated into letters by the usual A=1, B=2 code, give COFIN at the start and ANCED at the end, so the answer is COFINANCED.