Two professors, one of English and one of mathematics, were having drinks in the faculty club bar. “It is curious,” said the English professor, “how some poets can write one immortal line and nothing else of lasting value. John William Burgon, for example. His poems are so mediocre that no one reads them now, yet he wrote one of the most marvelous lines in English poetry: ‘A rose-red city half as old as Time.’”
The mathematician, who liked to annoy friends with improvised brainteasers, thought for a moment or two, then raised a glass and recited:
A rose-red city half as old as Time.
One billion years ago the city’s age
Was just two fifths of what Time’s age will be
A billion years from now. Can you compute
How old the crimson city is today?
The English professor had long ago forgotten algebra and quickly shifted the conversation to another topic, but readers of this department should have no difficulty with the problem.
The rose-red city’s age is seven billion years. Let x be the city’s present age and y be the present age of Time. A billion years ago the city would have been x – 1 billion years old, and a billion years from now Time’s age will be y + 1. The data in the problem permit two simple equations:
2x = y
x – 1 = 2⁄5 (y + 1)
These equations give x, the city’s present age, a value of seven billion years and y, Time’s present age, a value of 14 billion years.
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A version of this puzzle originally appeared in the October 1960 issue of Scientific American.