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Bob's Cab (Posted on 2009-03-02) Difficulty: 3 of 5
Cab driver Bob mentioned to his friend Jim that he recently drove three passengers in his cab, that the product of their (the passengers') ages was 2450, and that the sum of their ages was exactly twice Jim's age.

From this, Jim couldn't deduce what their three ages were.

But when Bob added that he was younger than at least one of the passengers, Jim, who knew Bob's age, was able to deduce all the passengers' ages.

What were Bob's and Jim's ages, and the ages of the passengers?

  Submitted by Charlie    
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Solution: (Hide)
2450 = 2 * 5 * 5 * 7 * 7

The following program produces all sets of three integers that multiply out to 2450 where no integer exceeds 120:

CLS
FOR twos1 = 0 TO 1
 rem2s = 1 - twos1
FOR fives1 = 0 TO 2
 rem5s = 2 - fives1
FOR sevens1 = 0 TO 2
 rem7s = 2 - sevens1

 FOR twos2 = 0 TO rem2s
 FOR fives2 = 0 TO rem5s
 FOR sevens2 = 0 TO rem7s

  a = INT(2 ^ twos1 * 5 ^ fives1 * 7 ^ sevens1 + .5)
  b = INT(2 ^ twos2 * 5 ^ fives2 * 7 ^ sevens2 + .5)
  c = 2450 / (a * b)
  IF a <= b AND b <= c AND c <= 120 THEN
    PRINT a; b; c, (a + b + c) / 2
  END IF

 NEXT
 NEXT
 NEXT

NEXT
NEXT
NEXT

The list, sorted by (a+b+c)/2 (Jim's age) and reformatted is:

a   b   c  (a+b+c)/2
           (Jim's age)
7  14  25     23
7  10  35     26
5  14  35     27
7   7  50     32
5  10  49     32
2  35  35     36
2  25  49     38
5   7  70     41
1  49  50     50
1  35  70     53
5   5  98     54
1  25  98     62

Only if Jim is 32 would there be ambiguity when Jim is told the sum of the ages.

In order for the ambiguity to be resolved by there being a passenger older than Bob, the passenger's ages would have to be 7, 7 and 50, with Bob's age being 49.

From Giant Book of Mensa Mind Challenges, by Todd Johnson, "Nearly Impossible Brain Bafflers" chapter, page 191.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsnow he knowsAdy TZIDON2009-03-07 10:05:24
SolutionSolutionDej Mar2009-03-02 17:52:14
assumptionsed bottemiller2009-03-02 13:09:07
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