Math Kangaroo USA
International Competition in Mathematics
for K-12 students

Login  |    Registration  |    Donate

Equal Sums 3

Equal Sums 3

Hints:

  • The total of the numbers in each circle is the same. Try adding all four circles together.

  • When you sum all four circles, you get the total of all the numbers (36) plus the four overlap regions (since each shared number is counted more than once).

  • Therefore, each individual circle adds up to:
    Total of all numbers (36)+Sum of overlaps÷4\text{Total of all numbers (36)} + \text{Sum of overlaps} \div 4
    That is: 9 + (Overlap Sum ÷ 4)

  • The smallest possible overlap sum is: 1 + 2 + 3 + 4 = 10
    The largest is: 5 + 6 + 7 + 8 = 26
    But it must be divisible by 4. So the valid overlap sums are:
    12, 16, 20, or 24

  • That means each circle’s total could be:
    9 + 3 = 12,
    9 + 4 = 13,
    9 + 5 = 14, or
    9 + 6 = 15

  • This narrows down the possibilities and gives you a solid place to start!