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Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

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Marbles

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

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More Marbles

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Strange Numbers

Stage: 3 Challenge Level: Challenge Level:1

We define "strange" numbers in the following way:

  • All strange numbers are prime.
     
  • Every single digit prime number is strange. 
     
  • A number with two or more digits is strange if, and only if, the two numbers obtained from it, by removing either its first or its last digit, are also strange.

Find all strange numbers.

This problem comes from the Hungarian magazine KoMaL. You can find many other challenging problems in English on the KoMaL website.


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