Problems & Puzzles: Puzzles

Puzzle 236. A Broken Face

Break a modern(*) Roman-numeral clock face into K pieces such that the sum of the resulting numbers on each piece are distinct prime numbers (p1, p2, ...pK) . Let's call Z to the sum of all these primes, and let's ask for solutions such that Z & K are maximum quantities (All numerals can be read from the center of the clock face.)

Old roman-numeral clock face

Q1. What are K & Z? (Show the distinct prime numbers thus generated)

Q2. Regarding the primes, the solution obtained, is unique, or are there other solutions?

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(*)
modern  means here that "4" is "IV" and not "IIII" as in the old clock's face shown above. BTW, this is a variation of a non-prime puzzle recently posted somewhere out there.

Solution:

Jon Wharf sent ten solutions. Both came with Z=82 and K=7.

(2, 3, 5, 7, 11, 13, 41)
(2, 3, 5, 7, 11, 17, 37)
(2, 3, 5, 7, 11, 23, 31)
(2, 3, 5, 11, 13, 17, 31)
(2, 3, 5, 7, 13, 23, 29)
(2, 3, 5, 7, 17, 19, 29)
(2, 3, 5, 11, 13, 19, 29)
(2, 3, 7, 11, 13, 17, 29)
(2, 3, 5, 13, 17, 19, 23)
(2, 3, 7, 11, 17, 19, 23)

All of these can be achieved by breaking the clock face (always splitting IV and IX). The partition with max 41 and the two with max 31 can be achieved with each prime on a continuous section of the circumference. In these cases the max prime starts in the middle of the IX and uses up the bulk of the X's

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