Sweets in jars

Number Theory Level 3

Sam has put sweets in five jars in such a way that no jar is empty and no two jars contain the same number of sweets. Also, any three jars contain more sweets in total that the total of the remaining two jars. What is the smallest possible number of sweets altogether in the five jars?


The answer is 35.

Julian Uy by Julian Uy , 26, Philippines

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1 solution

  • We can propose that: ( x − 2 ) + ( x − 1 ) + ( x ) > ( x + 1 ) + ( x + 2 ) (x - 2) + (x - 1) + (x) > (x + 1) + (x + 2) 3 x − 3 > 2 x + 3 3x - 3 > 2x + 3 x > 6 x > 6 .
  • Obviously, all jars have an integer number of sweets.
  • According to this, the least x we can afford is x = 7 x = 7 , because 7 is the closest integer greater than 6.
  • Replacing 7, we have that:
    Sum of all sweets is ( 7 − 2 ) + ( 7 − 1 ) + 7 + ( 7 + 1 ) + ( 7 + 2 ) = > 5 + 6 + 7 + 8 + 9 = 35 (7 - 2) + (7 - 1) + 7 + (7 + 1) + (7 + 2) => 5 + 6 + 7 + 8 + 9 = 35 .
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