Henry is eating breakfast in the kitchen of his spacious uptown flat, watching a blue jay resting on a branch of the cherry tree outside his window. A mockingbird sits in another branch of the tree, and when the blue jay sings, the mockingbird repeats the song. Henry believes the two birds to be singing exactly alike, but there is one difference that he does not notice: every fourth note the blue jay sings, the mockingbird reproduces twice, very quickly.
In a sequence beginning with the blue jay singing one note and the mockingbird producing the note twice, how many more notes than the blue jay will the mockingbird have produced, after the blue jay has sung 17 notes?
When Henry leaves his apartment to go to his office in the financial district, he finds a note on the windshield of his Saab indicating that a madman has kidnapped his fiancée, Claire, and is holding her in a secret lair somewhere in the city.
The madman is prematurely bald, like Henry's younger brother, Frederick. He is jealous of the couple's happiness, like Claire's landlord, Pietro, and an admirer of Claire's great beauty, like Henry's Uncle Samuel. Dr. Paul is head of surgery in the hospital where Uncle Samuel works as a male nurse; mocking nurses sometimes call him "Dr. Horseshoe," and joke behind Dr. Paul's back about the blinding gleam of hairless pink scalp beyond his forehead. Frederick leads a support group for other bald men that meets each Wednesday evening in the hospital cafeteria. One of the men in the group, _______, knows Uncle Samuel and has come to envy Henry, and to admire the subtle, forceful beauty of Claire, whom he has seen in photographs.
Who is the madman?
The madman begins to assume Henry's identity. He grows a handlebar mustache like Henry's, and wears the same cardigans and wide-waled corduroy pants that Henry favors. He begins peppering his speech with phrases Henry uses: "for the nonce," "moreover," and "mayhaps." Each time he adopts another aspect of Henry's identity the madman visits Claire in the basement lair, trying to convince her that it is he, Henry, come to rescue her. The madman can only adopt one characteristic per day, however, and by his careful estimation there are 27 characteristics he must adopt to pass himself off as Henry.
Meanwhile, Henry tries to discover the location of the madman's secret lair. From various clues left by the madman in his note, Henry knows that the lair is a) underground and b) within one mile of the hospital. By Henry's count, there are 14 locations that fit both categories. However, Henry's office is busy at this time of year—it is getting late in the tax season—and the building where he works is in the financial district, across town from the hospital, and so Henry is only able to investigate a secret lair every other day.
a) If only the last secret lair that Henry investigates will prove to be the one where Claire is being held, and if Henry investigates only one lair every other day, beginning on the first day of Claire's captivity, how many days will it take Henry to find the secret underground lair and rescue Claire?
b) If the madman can adopt only one of Henry's characteristics each day, beginning on the first day of Claire's captivity, how long will it take him to adopt all 27 characteristics?
c) Which will occur first: Henry finding Claire in the madman's secret lair, or the madman successfully adopting all of Henry's characteristics?
Claire's accommodations in the madman's lair are comfortable and spacious. The sub-basement of the building has been refurbished, and the space is heated, so that it is in fact warmer and nicer than Claire's own apartment, across town in the factory district. Even in that rundown section of the city, Claire's rent, $400 per month, is a strain on her monthly income of $850. For several months during the fall, Claire dropped hints to Henry that it would be economical for her to move in with him. Henry did not pick up on any of the hints, however, and when she asked him directly, Henry answered that his apartment was not large enough and, moreover, that studies showed those couples who cohabitated prior to marriage experienced higher incidences of divorce.
After 26 days, Claire has grown comfortable in the secret lair, which has an overstuffed couch and modest-sized television—not gaudily large, like Henry's—a queen-sized bed with 500 thread-count sheets and pillowcases, and, in the main sitting room, tasteful sea-green drapes hung over rectangles of blank wall.
Claire understands that the madman's aim is to convince her that he is Henry, come to save her. She at first considered the madman's ruse horrible, then futile, then silly. Now she sees it as strangely endearing. The madman has captured Henry's appearance, and replicated the slow, stilted movement of Henry's speech, the way that Henry sometimes looks over a person's shoulder rather than into her eyes, the pet names he has for her (Petunia, Pork Pie), the way that Henry will suddenly, after months or a year of abstinence, buy and smoke an entire pack of cigarettes and turn up acting strangely and reeking of tobacco.
But the madman, when he appeared today, was still bald. Has he overlooked the obvious—Henry's full head of curly dark hair? It's seemed so, the past few weeks, as his appearance has crept ever closer to Henry's while that glaring, horseshoe-shaped patch of scalp has remained. But the madman has been so exact in his imitations of Henry, and so thoughtful in furnishing the secret lair, that Claire thinks it must be something else.
Even so, Claire believes that even if he were to appear tomorrow with Henry's exact hair, the madman would still be distinguishable from Henry by the quality of ardor that Claire senses each time he bursts into the secret lair, announcing that he has come to deliver her from this captivity. The madman, clearly, does not know Henry personally. If he did he would know to disguise such passion (if, indeed, Henry feels such emotions at all). And he would hide, too, the show of chagrin he makes when Claire only looks at him stolidly and shakes her head.
The madman appeared today drinking peach brandy in an oversized snifter, as Henry often does. Since she rebuffed the madman a curious feeling has been growing in Claire, an interesting thought she has turned over while leafing through the magazines the madman has provided. Perhaps the madman imagines that she will fall in love with him—with the madman himself, and not merely a simulated Henry. Perhaps the madman does grasp Henry's true nature after all, and is ingeniously demonstrating what he, the madman, considers an improvement on Henry's cold taciturnity. And perhaps his bald spot, a great pink cul de sac that flushes purple each time Claire denies him, is a sign of himself that the madman is reluctant to conceal. Perhaps covering his baldness will be the last adjustment the madman makes, a final concealment that he would like to think won't be necessary.
Claire finds the notion charming, and even feels a flicker of something like tenderness when she thinks of the madman nursing this fragile hope. As the clock on the far wall turns toward night, and heat rustles in the vent, Claire wonders at the true nature of the madman. Is he a gentle man, moved to desperation? Perhaps he is not mad at all. Claire wonders if the madman will be bald when he appears again tomorrow.
The next day, when a man who looks like Henry bursts into the secret lair, shouting, Here I am, I've come to save you, darling, who is it? And what does Claire say?
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