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Summer Maths Puzzles - Thursday 8 August - 8 Octopuses

This puzzle, also taken from the first part from one of IBM’s Ponder This collection, seemed particularly apt for today.

An octopus has 8 arms which we can number from 1 to 8. 

We have eight octopuses named A,B,C,D,E,F,G and H.  Each octopus has a watch on each of its arms set to the times shown below.  These are special watches which show only the hour. 

  • Octopus A's watches on its 8 arms numbered 1..8 are set to 1, 2, 3, 4, 5, 6, 7 and 8 respectively.
  • Octopus B's watches on its 8 arms numbered 1..8 are set to 2, 4, 6, 8, 10, 12, 2 and 4 respectively.
  • October C's watches on its 8 arms numbered 1..8 are set to 3, 6, 9, 12, 3, 6, 9 and 12 respectively.
  • Octopus D's watches on its 8 arms numbered 1..8 are set to 4, 8, 12, 4, 8, 12, 4, 8 and 12 respectively.
  • Octopus E's watches on its 8 arms numbered 1..8 are set to 5, 10, 3, 8, 1, 6, 11 and 4 respectively.
  • ...and so on....

To adjust the hour on the watches, we can move all the watches of a single octopus (octopus C for example) one hour ahead, or move all the watches on a specific arm of all the octopuses (arm #3 for example) one hour ahead.

What adjustments do you need to make to fix it so that over 2/3 of the 64 watches show either 3, 6, 9 or 12?

 

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons