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Professor Matteossian's Memory

The following was supposed to be a humorous anecdote, making fun of mathematicians... and perhaps also mimicking an actual story about the great mathematician Ramanujan and G.H. Hardy. Rather than focus on the humor (if there is any), try to solve the problem....

Policeman (to bespectacled fat professor who has witnessed a smash): You say you saw the accident, sir. What was the number of the car that knocked this man down?

Professor Matteossian: I'm afraid I've forgotten it. But I remember noticing that if it were multiplied by itself, the cube root of the product would be equal to the sum of the digits reversed.

Assume: The car's license plate involves some combination of three letters and four digits.

Your task: What is(are) the possible number(s) on the license plate? And, why is the wording of the mathematician's response in the story made more complicated than necessary?


Source: M.D. Baughman's Educator's Handbook of Stories, Quotes, and Humor, 1963, p. 67

Hint: Play with some numbers and look for patterns...when will the cube root of the square of a number be an integer?

Remember that the number 2 can be written on a license plate as 0002.


Solution Commentary: One pattern is that for any number n, the cube root of n*n will be an integer if n is a cube itself. So, now, what are the possible n-values that can be written as a 4-digit sequence?

You may have felt that the last sentence is ambigious. That is, does the reversal of the digits refer to the original number or to the squared product? Thus, To make the problem perhaps harder, try to solve it using this latter interpretation as well...