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Tricky Tournament Treat (Posted on 2009-11-30) Difficulty: 3 of 5
Four friends, Arnold, Brian, Chuck and Denis, just played four rounds of a golf tournament. Their scores were all tied, even though the individual scores for the 16 rounds were all different. It is further known that:

(i) The 16 scores were all in the 60s and 70s.

(ii) All four of Arnold's scores were a prime.

(iii) All four of Brian's scores were semiprimes.

(iv) None of Chuck's or Denis' scores was either prime or semiprime.

(v) Chuck's lowest round was better (lower) than Denis’ lowest round. Chuck's worst round was better than Denis’ worst round.

What was the score of each of the golfers for the four rounds?

See The Solution Submitted by K Sengupta    
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Solution computer-assisted solution | Comment 2 of 3 |

The primes, semiprimes and others in the range are:

   prime semiprime  others
                     60
      61
           62
                     63
                     64
           65
                     66
      67
                     68
           69
                     70
      71
                     72
      73
           74
                     75
                     76
           77
                     78
      79
    ----  ---
     351  347

As there are 5 primes and 5 semiprimes, we need to remove one of each. The totals differ by 4 and so the prime to be removed must be 4 higher than the semiprime removed. This is possible only with 73 and 69 being the numbers removed. The total of each set is then 347-69 = 351-73 = 278.

The other two players' scores must be chosen from the "others" column. The sets of 4 chosen from this column that add up to 278 are:

 60  64  76  78
 60  68  72  78
 60  70  72  76
 63  64  75  76
 63  68  72  75
 64  66  70  78
 64  66  72  76
 64  68  70  76
 


The only pair of rows that don't share any number and where one of the pair has both the lower lowest number and also the lower highest number are:

 63  68  72  75
 64  66  70  78
 
So:
Arnold (prime):    61 67 71 79
Brian (semiprime): 62 65 74 77
Chuck:             63 68 72 75
Denis:             64 66 70 78

For first table:

   10   for I=60 to 79
   15     if prmdiv(I)=I then print I:PrTot=PrTot+I
   20     if fnSemiPrime(I) then print "     ";I:SpTot=SpTot+I
   25     if prmdiv(I)<I and fnSemiPrime(I)=0 then print "               ";I
   30   next
   40   print PrTot,SpTot
  100 
  200   end
  300   
  400   fnSemiPrime(X)
  410   if prmdiv(X)=X then return(0)
  420   N=X//prmdiv(X)
  430   if prmdiv(N)=N then return(-1)
  440   return(0)

For second table:

DATA 60,63,64,66,68,70,72,75,76,78

FOR i = 1 TO 10: READ num(i): NEXT

FOR a = 1 TO 7
FOR b = a + 1 TO 8
FOR c = b + 1 TO 9
FOR d = c + 1 TO 10
  IF num(a) + num(b) + num(c) + num(d) = 278 THEN
      PRINT num(a); num(b); num(c); num(d)
  END IF
NEXT
NEXT
NEXT
NEXT

 


  Posted by Charlie on 2009-11-30 13:59:39
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