MATH 135H Assignment #1

Due: 27 September, 2000

The following excerpt from an obscure story poses a mathematical problem. Your mission is to read it and answer the questions at the end.

"Yes, m'm, Master is at home, m'm," said the stately old butler. (N.B.---it is only a butler of experience who can manage a series of three M's together, without any interjacent vowels.) "And the ole party is a-waiting you in the libery."

"I don't like his calling your father an old party," Mad Mathesis whispered to her niece, as they crossed the hall. And Clara had only just time to whisper in reply "he meant the whole party," before they were ushered into the library, and the sight of the five solemn faces there assembled chilled her into silence.

Her father sat at the head of the table, and mutely signed to theladies to take the two vacant chairs, one on each side of him. His three sons and Balbus completed the party. Writing materials had been arranged round the table, after the fashion of a ghostly banquet: the butler had evidently bestowed much thought on the grim device. Sheets of quarto paper, each flanked by a pen on one side and a pencil on the other, represented the plates-penwipers did duty for rolls of bread---while ink-bottles stood in the places usually occupied by wine-glasses. The pièce de resistance was a large green baize bag, which gave forth, as the old man restlessly lifted it from side to side, a charming jingle, as of innumerable golden guineas.

"Sister, daughter, sons---and Balbus---," the old man began, so nervously, that Balbus put in a gentle "Hear, hear!" while Hugh drummed on the table with his fists. This disconcerted the unpractised orator. "Sister---" he began again, then paused a moment, moved the bag to the other side, and went on with a rush, "I mean---this being---a critical occasion---more or less-being the year when one of my sons comes of age---" he paused again in some confusion, having evidently got into the middle of his speech sooner than he intended: but it was too late to go back. "Hear, hear!" cried Balbus. "Quite so," said the old gentleman, recovering his self-possession a little: "when first I began this annual custom---my friend Balbus will correct me if I am wrong---" (Hugh whispered "with a strap!" but nobody heard him except Lambert, who only frowned and shook his head at him) "---this annual custom of giving each of my sons as many guineas as would represent his age---it was a critical time---so Balbus informed me---as the ages of two of you were together equal to that of the third-so on that occasion I made a speech---" He paused so long that Balbus thought it well to come to the rescue with the words "It was a most---" but the old man checked him with a warning look: "yes, made a speech," he repeated. "A few years after that, Balbus pointed out---I say pointed out---" ("Hear, hear!" cried Balbus. "Quite so," said the grateful old man.) "---that it was another critical occasion. The ages of two of you were double that of the third. So I made another speech---another speech. And now again it's a critical occasion---so Balbus says---and I am making---" (Here Mad Mathesis pointedly referred to her watch) "all the haste I can!" the old man cried, with wonderful presence of mind. "Indeed, sister, I'm coming to the point now! The number of years that have passed since that first occasion is just two-thirds of the number of guineas I then gave you. Now, my boys, calculate your ages from the data, and you shall have the money!"

"But we know our ages!" cried Hugh.

"Silence, sir!" thundered the old man, rising to his full height (he was exactly five-foot five) in his indignation. "I say you must use the data only! You mustn't even assume which it is that comes of age!" He clutched the bag as he spoke, and with tottering steps (it was about as much as he could do to carry it) he left the room.

"And you shall have a similar cadeau," the old lady whispered to her niece, "when you've calculated the percentage!" And she followed her brother.

Nothing could exceed the solemnity with which the old couple had risen from the table, and yet was it---was it a grin with which the father turned away from his unhappy sons? Could it be---could it be a wink with which the aunt abandoned her despairing niece? And were those---were those sounds of suppressed chuckling which floated into the room, just before Balbus (who had followed them out) closed the door? Surely not: and yet the butler told the cook---but no, that was merely idle gossip, and I will not repeat it.

The shades of evening granted their unuttered petition, and "closed not o'er" them (for the butler brought in the lamp): the same obliging shades left them a "lonely bark" (the wail of a dog, in the back-yard, baying at the moon) for "awhile": but neither "morn, alas" (nor any other epoch) seemed likely to "restore" them---to that peace of mind which had once been theirs ere ever these problems had swooped upon them, and crushed them with a load of unfathomable mystery!

"It's hardly fair," muttered Hugh, "to give us a jumble as this to work out!"

"Fair?" Clara echoed, bitterly. "Well!"

And to all my gentle readers I can but repeat the last words of gentle Clara---


1. State the mathematical problem being posed in your own words. [2]

2. Set up the mathematical problem as an equation or equations to be solved. [3]

3. Solve the mathematical problem. (With or without the equations from question 2.) Include a complete explanation of why your solution is correct. [5]

Bonus. Who wrote this tale? Where did it first appear? [1]

Solutions to Assignment #1
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