Brain Teasers
Gauss and an Old Professor
There was a famous mathematician named Karl Friedrich Gauss. When he was young, he entered Gotinga University. One day he met an old professor, and spoke to him. They chatted, and the professor asked Gauss about his age.
Gauss said: "The cube and the fourth power of my age contain each digit from 0 to 9 exactly once."
The professor said: "Very good. It's good coincidence because the square and the cube of my age contain each digit from 0 to 9 exactly once."
How old was Gauss, and how old was the professor?
Gauss said: "The cube and the fourth power of my age contain each digit from 0 to 9 exactly once."
The professor said: "Very good. It's good coincidence because the square and the cube of my age contain each digit from 0 to 9 exactly once."
How old was Gauss, and how old was the professor?
Hint
Eliminate the impossibilities, such as their ages can't end with 0.Answer
Their ages can't end with 0, 1, 5 or 6, and must be either a multiple of 3 or a multiple of 9 minus 1 because the sum of its square and its cube or its cube and its fourth power is divisible by 9.Gauss's age is easy. Since 17^3=4913, 17^4=83521, 22^3=10648, 22^4=234256, Gauss's age is between 18 and 21. The only possibility is 18. 18^3=5832, 18^4=104976.
The professor's age is a little harder. It can be 47 to 99, and the possibilities are: 48, 53, 54, 57, 62, 63, 69, 72, 78, 84, 87, 89, 93, 98, 99. By trial and error, 69 satisfies this condition: 69^2=4761, 69^3=328509.
Therefore, Gauss was 18 and the professor was 69.
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Comments
In the teaser, I thought you meant that each power of the ages had to contain all digits 0 - 9. I was wondering just how old these people were.
Same issue here. Can you rephrase this to make it somewhat clearer?
WOW
That was so confusing!!! At first I thought that you meant that each power had to contain all ten digits. When I realized that it was impossible to have people that old, I thought I was supposed to add the fourth power and cube of Gauss's age and the cube and square of the professor's age. I finally gave up and looked at the answer. You should be a little more specific.
Yup I had the same problem! And gave up and looked at the answer!
I understood the wording just fine, but, as always, was chilled by trial and error stuff that requires a spreadsheet for a reasonably swift answer.
This was just so hard. after 2 seconds i peeked at the answers!
Interesting trivia. I reduced the problem the same way you did, but then resorted to a spreadsheet to find the answer since there were still too many possible solutions.
I like the teaser even though I kind of agree with Stil's comment.
I like the teaser even though I kind of agree with Stil's comment.
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