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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 Solution Comment 3 of 3 |

Arnold: 61 + 67 + 71 + 79 = 278 
Brian : 62 + 65 + 74 + 77 = 278
Chuck : 63 + 68 + 72 + 75 = 278
Denis : 64 + 66 + 70 + 78 = 278

The score of each of the golfers for the four rounds was 278. 

The four round score can be deduced from the first three clues....(i), (ii) and (iii):
 Four distinct primes between 60 and 79
 {61, 67, 71, 73, 79}:
     61 + 67 + 71 + 73 = 272
     61 + 67 + 71 + 79 = 278 <---
     61 + 67 + 73 + 79 = 280
     61 + 71 + 73 + 79 = 284
     67 + 71 + 73 + 79 = 290
 Four distinct semiprimes betwen 60 and 79
 {62, 65, 69, 74, 77}:

     62 + 65 + 69 + 74 = 270
     62 + 65 + 69 + 77 = 273
     62 + 65 + 74 + 77 = 278 <---
     62 + 69 + 74 + 77 = 282
     65 + 69 + 74 + 77 = 285
The only common value between each of the sums of  four primes and four semiprimes is 278.

There are eight sets of non-prime, non-semiprime numbers inclusively between 60 and 79 that totals 278:
     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
Of these, there are only three pairs of sets where the numbers are distinct:
     60 + 64 + 76 + 78 <=> 63 + 68 + 72 + 75
     60 + 68 + 72 + 78 <=> 63 + 64 + 75 + 76
     63 + 68 + 72 + 75 <=> 64 + 66 + 70 + 78
Of these, only one pair of sets has the low and high values lower than the other:
     63 + 68 + 72 + 75 <=> 64 + 66 + 70 + 78

Edited on November 30, 2009, 6:37 pm
  Posted by Dej Mar on 2009-11-30 18:00:09

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