There is a safe with a 5 digit No. The 4th digit is 4 greater than second digit, while 3rd digit is 3 less than 2nd digit. The 1st digit is thrice the last digit. There are 3 pairs whose sum is 11. Find the number.

• First of all, I think the last sentence should be modified. In fact, saying "There are 3 pairs in the number whose sum is 11" means the number has a sum of 11 i.e. the sum of its digits is 11. IMHO saying "the number has 3 pairs whose sum is 11" would avoid confusion.
  
  Now for the solution.
  
  There are 5 digits, let's call them a,b,c,d,e
  
  The 4th digit is 4 greater than the second digit,
  
  d = b+4
  
  while the 3rd digit is 3 less than the 2nd digit.
  
  c = b-3
  
  The 1st digit is thrice the last digit.
  
  a = 3*e
  
  This means that there are only 2 variables, b and e, since all the others can be written in terms of b and e.
  
  b MUST be greater than or equal to 3, otherwise c will be negative
  
  b MUST be less than or equal to 5, otherwise d will be 10 or more (i.e. not a single digit)
  
  e is between 0 and 3. 4 or more would put a in the double digits.
  
  There are 10 possible pairs (Combination of 5-choose-2) of which 5 can never be 11;
  
  
  a+e = 3e+e = 4e can never be 11 if e is an integer
  
  b+c = 2b - 3 can't be more than 7
  
  b+d = 2b + 4 can't be 11 if b is an integer
  
  b+e can't be more than 8
  
  c+e = b+e-3 can't be more than 5
  
  Furthermore,
  
  a+ d can only be 11 if b=4 and e=1, and this makes it the onlt pair with a sum of 11.
  
  So we eliminate these 6 pairs, and we are left with 4 pairs, 3 of which must sum up to 11.
  
  The most obvious one is c + d because it only depends on b (c + d = b - 3 + b + 4 = 2b + 1)
  
  Solving for b we get b = 5
  
  the pair a + b = b + 3e
  solving a + b = 11 gives us e = 2
  
  with b = 5 and e = 2 the pair (d+e) will also sum up to 11 and thus we have our 3 pairs.
  
  a = 3e = 3*2 = 6
  b = 5
  c = b -3 = 2
  d = b +4 = 9
  e = 2
• 12 years agoHelpfull: Yes(9) No(0)
• 65292
• 14 years agoHelpfull: Yes(1) No(4)
• nice question
• 14 years agoHelpfull: Yes(1) No(6)